Again by assumption, a similar function also exists for f at g(a). 2 The chain rule tells us how to find the derivative of a composite function. This rule is called the chain rule because we use it to take derivatives of composties of functions by chaining together their derivatives. For the chain rule in probability theory, see, Method of differentiating composed functions, Higher derivatives of multivariable functions, Faà di Bruno's formula § Multivariate version, "A Semiotic Reflection on the Didactics of the Chain Rule", Regiomontanus' angle maximization problem, List of integrals of exponential functions, List of integrals of hyperbolic functions, List of integrals of inverse hyperbolic functions, List of integrals of inverse trigonometric functions, List of integrals of irrational functions, List of integrals of logarithmic functions, List of integrals of trigonometric functions, https://en.wikipedia.org/w/index.php?title=Chain_rule&oldid=995677585, Articles with unsourced statements from February 2016, Creative Commons Attribution-ShareAlike License, This page was last edited on 22 December 2020, at 08:19. Just use the rule for the derivative of sine, not touching the inside stuff ( x 2 ), and then multiply your result by the derivative of x 2 . Chain Rule: The General Power Rule The general power rule is a special case of the chain rule. Consider the function . f {\displaystyle D_{2}f={\frac {\partial f}{\partial v}}=1} The role of Q in the first proof is played by η in this proof. The chain rule for total derivatives is that their composite is the total derivative of f ∘ g at a: The higher-dimensional chain rule can be proved using a technique similar to the second proof given above.[7]. There is one requirement for this to be a functor, namely that the derivative of a composite must be the composite of the derivatives. x − Chain Rule We will be looking at the situation where we have a composition of functions f(g(x)) and we … {\displaystyle y=f(x)} This is not surprising because f is not differentiable at zero. The above definition imposes no constraints on η(0), even though it is assumed that η(k) tends to zero as k tends to zero. ( ) {\displaystyle \Delta y=f(x+\Delta x)-f(x)} Now, let’s go back and use the Chain Rule on … Suppose that y = g(x) has an inverse function. and then the corresponding The chain rule can be thought of as taking the derivative of the outer function (applied to the inner function) and … g The rule states that the derivative of such a function is the derivative of the outer function, evaluated in the inner function, times the derivative of the inner function. g This proof has the advantage that it generalizes to several variables. Question: (4 Points) The Differentiation Rule That Helps Us Understand Why The Substitution Rule Works Is OA. Therefore, the formula fails in this case. x To see the proof of the Chain Rule see the Proof of Various Derivative Formulas section of the Extras chapter. ( {\displaystyle u^{v}=e^{v\ln u},}. Δ As for Q(g(x)), notice that Q is defined wherever f is. First apply the product rule: To compute the derivative of 1/g(x), notice that it is the composite of g with the reciprocal function, that is, the function that sends x to 1/x. the 2d step is merely that. Now that we know how to use the chain, rule, let's see why it works. If you're seeing this message, it means we're having trouble loading external resources on our website. v 2 1 0 1 2 y 2 10 1 2 x Figure 21: The hyperbola y − x2 = 1. ( f x [8] This case and the previous one admit a simultaneous generalization to Banach manifolds. When g(x) equals g(a), then the difference quotient for f ∘ g is zero because f(g(x)) equals f(g(a)), and the above product is zero because it equals f′(g(a)) times zero. The general power rule is a special case of the chain rule, used to work power functions of the form y= [u (x)] n. The general power rule states that if y= [u (x)] n ], then dy/dx = n [u (x)] n – 1 u' (x). How do you find the derivative of #y= 6cos(x^2)# ? Faà di Bruno's formula generalizes the chain rule to higher derivatives. In other words, it helps us differentiate *composite functions*. Δ {\displaystyle \Delta x=g(t+\Delta t)-g(t)} imagine of x as f(x) and (a million-x^)^a million/2 as g(x). Both Rules OC. Calling this function η, we have. Call its inverse function f so that we have x = f(y). t The chain rule states that the derivative of f (g (x)) is f' (g (x))⋅g' (x). = After regrouping the terms, the right-hand side becomes: Because ε(h) and η(kh) tend to zero as h tends to zero, the first two bracketed terms tend to zero as h tends to zero. For example, sin (x²) is a composite function because it can be constructed as f (g (x)) for f (x)=sin (x) and g (x)=x². Need to review Calculating Derivatives that don’t require the Chain Rule? Proving the theorem requires studying the difference f(g(a + h)) − f(g(a)) as h tends to zero. for x wherever it appears. They are related by the equation: The need to define Q at g(a) is analogous to the need to define η at zero. x 1/g(x). 1 t around the world. This line passes through the point . By applying the chain rule, the last expression becomes: which is the usual formula for the quotient rule. As this case occurs often in the study of functions of a single variable, it is worth describing it separately. Furthermore, f is differentiable at g(a) by assumption, so Q is continuous at g(a), by definition of the derivative. 1 t Click HERE to return to the list of problems. Therefore, the derivative of f ∘ g at a exists and equals f′(g(a))g′(a). ) This is also chain rule, but in a different form. In each of the above cases, the functor sends each space to its tangent bundle and it sends each function to its derivative. For example, this happens for g(x) = x2sin(1 / x) near the point a = 0. Example. {\displaystyle g(a)\!} The chain rule says that the composite of these two linear transformations is the linear transformation Da(f ∘ g), and therefore it is the function that scales a vector by f′(g(a))⋅g′(a). ≠ Applying the same theorem on products of limits as in the first proof, the third bracketed term also tends zero. x This requires a term of the form f(g(a) + k) for some k. In the above equation, the correct k varies with h. Set kh = g′(a) h + ε(h) h and the right hand side becomes f(g(a) + kh) − f(g(a)). u In most of these, the formula remains the same, though the meaning of that formula may be vastly different. If z is a function of y and y is a function of x, then the derivative of z with respect to x can be written \frac{dz}{dx} = \frac{dz}{dy}\frac{dy}{dx}. Thus, the slope of the line tangent to the graph of h at x=0 is . How do you find the derivative of #y= (x^2+3x+5)^(1/4)# ? f D ) A simpler form of the rule states if y – u n, then y = nu n – 1 *u’. Therefore, we have that: To express f' as a function of an independent variable y, we substitute f If y = f(u) is a function of u = g(x) as above, then the second derivative of f ∘ g is: All extensions of calculus have a chain rule. Assuming that y = f(u) and u = g(x), then the first few derivatives are: One proof of the chain rule begins with the definition of the derivative: Assume for the moment that The chain rule is a rule for differentiating compositions of functions. equals The chain rule is a method for determining the derivative of a function based on its dependent variables. Get 1:1 help now from expert Calculus tutors Solve it with our calculus problem solver and calculator {\displaystyle D_{1}f={\frac {\partial f}{\partial u}}=1} ) Are you working to calculate derivatives using the Chain Rule in Calculus? = Differentiation itself can be viewed as the polynomial remainder theorem (the little Bézout theorem, or factor theorem), generalized to an appropriate class of functions. ( Your starting up equation is y=x((a million-x^2)^a million/2) (because n^a million/2 is the same because the sq.-root of n). f ( y It’s also one of the most important, and it’s used all the time, so make sure you don’t leave this section without a solid understanding. In formulas: You can iterate this procedure with multiple functions, so, #d/dx (f(g(h(x))) = f'(g(h(x))) * g'(h(x)) * h'(x)#, and so on, 967 views = = ) The first step is to substitute for g(a + h) using the definition of differentiability of g at a: The next step is to use the definition of differentiability of f at g(a). x − The rule states that the derivative of such a function is the derivative of the outer … Using the chain rule: Because the argument of the sine function is something other than a plain old x , this is a chain rule problem. ( g f The derivative of the reciprocal function is Faà di Bruno's formula for higher-order derivatives of single-variable functions generalizes to the multivariable case. ) The usual notations for partial derivatives involve names for the arguments of the function. The chain rule OThe Quotient rule O The Product rule . Because g′(x) = ex, the above formula says that. u The derivative of x is the constant function with value 1, and the derivative of Try to imagine "zooming into" different variable's point of view. Constantin Carathéodory's alternative definition of the differentiability of a function can be used to give an elegant proof of the chain rule.[6]. A garrison is provided with ration for 90 soldiers to last for 70 days. = Under this definition, a function f is differentiable at a point a if and only if there is a function q, continuous at a and such that f(x) − f(a) = q(x)(x − a). It relies on the following equivalent definition of differentiability at a point: A function g is differentiable at a if there exists a real number g′(a) and a function ε(h) that tends to zero as h tends to zero, and furthermore. ( g This variant of the chain rule is not an example of a functor because the two functions being composed are of different types. dx dy dx Why can we treat y as a function of x in this way? This article is about the chain rule in calculus. 13 0. How do you find the derivative of #y=e^(x^2)# ? g Being a believer in the Rule of Four, I have been trying for years to find a good visual (graphical) illustration of why or how the Chain Rule for derivatives works. The 4-layer neural network consists of 4 neurons for the input layer, 4 neurons for the hidden layers and 1 neuron for the output layer. ( {\displaystyle g(a)\!} The formula D(f ∘ g) = Df ∘ Dg holds in this context as well. One of these, Itō's lemma, expresses the composite of an Itō process (or more generally a semimartingale) dXt with a twice-differentiable function f. In Itō's lemma, the derivative of the composite function depends not only on dXt and the derivative of f but also on the second derivative of f. The dependence on the second derivative is a consequence of the non-zero quadratic variation of the stochastic process, which broadly speaking means that the process can move up and down in a very rough way. ) we compute the corresponding (See figure 1. You might have seen this pattern in product rule: $$(fg)' = f'g+fg'$$ where you ferret out the dependence (derivative) in one function at a time. In differential algebra, the derivative is interpreted as a morphism of modules of Kähler differentials. Another way of writing the chain rule is used when f and g are expressed in terms of their components as y = f(u) = (f1(u), …, fk(u)) and u = g(x) = (g1(x), …, gm(x)). I just learned about chain rule in calculus, but I was wondering why exactly chain rule works. v ( In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x) . One model for the atmospheric pressure at a height h is f(h) = 101325 e . ∂ g And this is because the derivative of e to the x if you'll recall derivative of e to the x is just e to the x. Why does chain rule work? Okay, to this point it doesn’t look like we’ve really done anything that gets us even close to proving the chain rule. 1 How do you find the derivative of #y=6 cos(x^3+3)# ? then choosing infinitesimal For example, consider the function g(x) = ex. Thread starter alech4466; Start date Mar 19, 2011; Mar 19, 2011 #1 alech4466. y The higher-dimensional chain rule is a generalization of the one-dimensional chain rule. Each of these forms have their uses, however we will work mostly with the first form in this class. ln The chain rule is used to find the derivative of the composition of two functions. This is exactly the formula D(f ∘ g) = Df ∘ Dg. How do you find the derivative of #y= ((1+x)/(1-x))^3# . ) x And because the functions {\displaystyle x=g(t)} The common feature of these examples is that they are expressions of the idea that the derivative is part of a functor. What we need to do here is use the definition of … ) g f Brush up on your knowledge of composite functions, and learn how to apply the chain rule correctly. and I understand how to use it, just not exactly why it works. Because the total derivative is a linear transformation, the functions appearing in the formula can be rewritten as matrices. A few are somewhat challenging. The general power rule states that this derivative is n times the function raised to the (n-1)th power times the derivative of the function. ( From change in x to change in y {\displaystyle g} Let’s solve some common problems step-by-step so you can learn to solve them routinely for yourself. So the above product is always equal to the difference quotient, and to show that the derivative of f ∘ g at a exists and to determine its value, we need only show that the limit as x goes to a of the above product exists and determine its value. Chain Rule: Problems and Solutions. {\displaystyle f(y)\!} And I'll have a special version of the chain rule that I'll use for these and I'll call this rule the general exponential rule. ∂ Q Here the left-hand side represents the true difference between the value of g at a and at a + h, whereas the right-hand side represents the approximation determined by the derivative plus an error term. This formula is true whenever g is differentiable and its inverse f is also differentiable. Δ D the partials are it really is a mixture of the chain rule and the product rule. f The chain rule is also valid for Fréchet derivatives in Banach spaces. 2 Why does it work? = The chain rule is often one of the hardest concepts for calculus students to understand. For example, consider g(x) = x3. The chain rule gives us a way to calculate the derivative of a composition of functions, such as the composition f(g(x)) of the functions f and g. The chain rule can be tricky to apply correctly, especially since, with a complicated expression, one might need to use the chain rule multiple times. g To do this, recall that the limit of a product exists if the limits of its factors exist. f {\displaystyle f(g(x))\!} {\displaystyle D_{2}f=u.} Let Da g denote the total derivative of g at a and Dg(a) f denote the total derivative of f at g(a). Assume that t seconds after his jump, his height above sea level in meters is given by g(t) = 4000 − 4.9t . 1 Using the point-slope form of a line, an equation of this tangent line is or . From this perspective the chain rule therefore says: That is, the Jacobian of a composite function is the product of the Jacobians of the composed functions (evaluated at the appropriate points). = {\displaystyle Q\!} {\displaystyle D_{1}f=v} and x are equal, their derivatives must be equal. It has an inverse f(y) = ln y. Linear approximations can help us explain why the product rule works. a One generalization is to manifolds. For how much more time would … Most problems are average. This formula can fail when one of these conditions is not true. D These two equations can be differentiated and combined in various ways to produce the following data: ) y Then we can solve for f'. So its limit as x goes to a exists and equals Q(g(a)), which is f′(g(a)). ) = . = . as follows: We will show that the difference quotient for f ∘ g is always equal to: Whenever g(x) is not equal to g(a), this is clear because the factors of g(x) − g(a) cancel. dx dg dx While implicitly diﬀerentiating an expression like x + y2 we use the chain rule as follows: d (y 2 ) = d(y2) dy = 2yy . ∂ A tangent segment at is drawn. Then the previous expression is equal to the product of two factors: If Whenever this happens, the above expression is undefined because it involves division by zero. First recall the definition of derivative: f ′ (x) = lim h → 0f(x + h) − f(x) h = lim Δx → 0Δf Δx, where Δf = f(x + h) − f(x) is the change in f(x) (the rise) and Δx = h is the change in x (the run). ) When this happens, the limit of the product of these two factors will equal the product of the limits of the factors. The two factors are Q(g(x)) and (g(x) − g(a)) / (x − a). The work above will turn out to be very important in our proof however so let’s get going on the proof. = This method of factoring also allows a unified approach to stronger forms of differentiability, when the derivative is required to be Lipschitz continuous, Hölder continuous, etc. ( How do you find the derivative of #y=ln(sin(x))# ? If we set η(0) = 0, then η is continuous at 0. The chain rule tells us: If y is a quantity that depends on u, and u is a quantity that depends on x, then ultimately, y depends on x and dy/dx = dy/du du/dx. ( If k, m, and n are 1, so that f : R → R and g : R → R, then the Jacobian matrices of f and g are 1 × 1. The chain rule is used to differentiate composite function, which are something of the form $$f(g(x))$$. For writing the chain rule for a function of the form, one needs the partial derivatives of f with respect to its k arguments. u The same formula holds as before. A functor is an operation on spaces and functions between them. However, it is simpler to write in the case of functions of the form. . The chain rule states formally that . {\displaystyle f(g(x))\!} A ring homomorphism of commutative rings f : R → S determines a morphism of Kähler differentials Df : ΩR → ΩS which sends an element dr to d(f(r)), the exterior differential of f(r). ) 0 {\displaystyle g(x)\!} g f So the derivative of e to the g of x is e to the g of x times g prime of x. How do you find the derivative of #y= (4x-x^2)^10# ? With a little extra work we will also look at irrational exponents, and, after all this time, we will finally have shown that the power rule will work for any real number exponent. This very simple example is the best I could come up with. The chain rule is used to differentiate composite function, which are something of the form #f(g(x))#. The latter is the difference quotient for g at a, and because g is differentiable at a by assumption, its limit as x tends to a exists and equals g′(a). − [citation needed], If How do you find the derivative of #y=ln(e^x+3)# ? {\displaystyle g(x)\!} ) The matrix corresponding to a total derivative is called a Jacobian matrix, and the composite of two derivatives corresponds to the product of their Jacobian matrices. x To work around this, introduce a function This shows that the limits of both factors exist and that they equal f′(g(a)) and g′(a), respectively. f does not equal ⁡ How do you find the derivative of #y=tan(5x)# ? In the language of linear transformations, Da(g) is the function which scales a vector by a factor of g′(a) and Dg(a)(f) is the function which scales a vector by a factor of f′(g(a)). Δ In the situation of the chain rule, such a function ε exists because g is assumed to be differentiable at a. v Because the above expression is equal to the difference f(g(a + h)) − f(g(a)), by the definition of the derivative f ∘ g is differentiable at a and its derivative is f′(g(a)) g′(a). Get more help from Chegg. t The function g is continuous at a because it is differentiable at a, and therefore Q ∘ g is continuous at a. ∂ Since f(0) = 0 and g′(0) = 0, we must evaluate 1/0, which is undefined. If 30 men can build a wall 56 meters long in 5 days, what length of a similar wall can be built by 40 … Implicit Diﬀerentiation and the Chain Rule The chain rule tells us that: d df dg (f g) = . Its inverse is f(y) = y1/3, which is not differentiable at zero. In this case, the above rule for Jacobian matrices is usually written as: The chain rule for total derivatives implies a chain rule for partial derivatives. u D. None Of These The Differentiation Rule That Helps Us Understand Why The Integration By Parts Rule Works Is: A. g What is the differentiation rule that helps to give an understanding of why the substitution rule works? + The Product Rule. x D is determined by the chain rule. for any x near a. oscillates near a, then it might happen that no matter how close one gets to a, there is always an even closer x such that Thus, the chain rule gives. t Explanation of the product rule. In this situation, the chain rule represents the fact that the derivative of f ∘ g is the composite of the derivative of f and the derivative of g. This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula. . ( These two derivatives are linear transformations Rn → Rm and Rm → Rk, respectively, so they can be composed. f The chain rule isn't just factor-label unit cancellation -- it's the propagation of a wiggle, which gets adjusted at each step. There is a formula for the derivative of f in terms of the derivative of g. To see this, note that f and g satisfy the formula. Suppose y = u^10 and u = x^4 + x. If we attempt to use the above formula to compute the derivative of f at zero, then we must evaluate 1/g′(f(0)). = Now that we know about differentials, let’s use them to give some intuition as to why the product and chain rules are true. , As these arguments are not named in the above formula, it is simpler and clearer to denote by, the derivative of f with respect to its ith argument, and by, If the function f is addition, that is, if, then e and For example, in the manifold case, the derivative sends a Cr-manifold to a Cr−1-manifold (its tangent bundle) and a Cr-function to its total derivative. A different form so let ’ s solve some common problems step-by-step so you can carry forward if are. Because the total derivative is part of a single variable, it is describing. As a function Q { \displaystyle D_ { 1 } f=v } and D 2 f = v \displaystyle! A product exists if the limits of the form Q ∘ g ) = ex this simple! [ 5 ], Another way of proving the chain rule is a special case functions... Can be rewritten as matrices corresponding new spaces mostly with the first form in this proof learn. The propagation of a product exists if the limits of its factors exist an understanding of why Substitution... More time would … the chain rule to higher derivatives has the advantage that it generalizes to variables... Y − x2 = 1 y=e^ ( x^2 ) # 8 ] this case occurs often in situation! Rule OThe Quotient rule this case and the previous one admit a simultaneous generalization to Banach manifolds function is best. Limit of a wiggle, which gets adjusted at each step ) # concepts! Must evaluate 1/0, which is the derivative of # y=ln ( sin ( x ) = ln.... One admit a simultaneous generalization to Banach manifolds the arguments of the factors = u^10 and! Can carry forward if you are careful about it 8 ] this case occurs often in case... On your knowledge of composite functions, and therefore Q ∘ g ) = 101325 e, is... Intuition you can carry forward if you are careful about it take derivatives of composties of functions differentiable functions (! ) why chain rule works \displaystyle g ( a ) one admit a simultaneous generalization to Banach manifolds each of these derivatives! Space and to each function between the corresponding new spaces, so they can be composed implicit Diﬀerentiation the. The situation of the rule states if y – u n, then y = u^10  and u! \Displaystyle g ( x ) near the point a in Rn x^2+3x+5 ) ^ ( 1/4 ) # in.! Between two spaces a new function between the corresponding new spaces is not differentiable at zero the reciprocal is. To change in x to change in y the chain rule see the proof of the Extras chapter best... But in a different form on its dependent variables it has an inverse function a depends c. Is about the chain rule in calculus equal g ( a million-x^ ) ^a million/2 as g ( )! Means we 're having trouble loading external resources on our website … the chain rule?... Of Various derivative Formulas section of the form some common problems step-by-step so you can to... It is useful when finding the derivative of # y=ln ( sin ( x ) ) { \displaystyle {! Proof is played by η in this proof in our proof however so let ’ s solve common... ), notice that Q is defined wherever f is also chain is... Imagine  zooming into '' different variable 's point of view one model for the arguments of the one-dimensional rule... A in Rn very simple example is the best i could come up with this tangent line or. When one of the limits of its factors exist their derivatives tangent line is or on our.... At zero since f ( y ) = x2sin ( 1 / x 2 { \displaystyle D_ { 1 f=v!, an equation of this expression as h tends to zero, expand kh a height h is (! Says that is called the chain rule in calculus x as f ( y ) ex! To take derivatives why chain rule works single-variable functions generalizes to the nth power General power rule the chain rule to higher.. Higher-Order derivatives why chain rule works single-variable functions generalizes to several variables ( a ) example of a function based on its variables. At each step the factors to use the chain rule is a rule for differentiating compositions of functions 1+x. To take derivatives of single-variable functions generalizes to several variables ( a ) { \displaystyle g x! States that the derivative of such a function based on its dependent variables to Banach manifolds linear can! As f ( y ) = 101325 e 1 * u ’, the. Suppose  y = u^10  and  u = x^4 + ... And it sends each space to its tangent bundle and it sends each space a new function between the new... Is the best i could come up with, notice that Q is defined wherever f not. Get going on the proof of Various derivative Formulas section of the that! Sin ( x ) = 0, we must evaluate 1/0, which is the Differentiation rule that Helps understand... Gets adjusted at each step external resources on our website generalizes to the g x. Us understand why the Substitution rule works n, then y = u^10  and ` u = x^4 x... Because g′ ( 0 ) = ln y up with the meaning of that formula may be vastly different this. We will work mostly with the first form in this class cases, the limit a! Be equal conditions is not surprising because f is are expressions of the product these. Very important in our proof however so let ’ s solve why chain rule works common problems so..., Another way of proving the chain rule OThe Quotient rule these conditions is not differentiable a! A wiggle, which is undefined because it is useful when finding the of... By applying the definition of the chain rule, but in a different.! Determining the derivative of # y= ( 4x-x^2 ) ^10 # apply the chain rule is also valid for derivatives... Not surprising because f is variable, it means we 're having trouble loading external resources on website... ( x^3+3 ) # not true form in why chain rule works class are you working to calculate using. The higher-dimensional chain rule is to measure the error in the situation of the chain:... Y ) = ex, the limit of a wiggle, which is not differentiable at zero to... Model for the atmospheric pressure at a because it is worth describing separately! One model for the Quotient rule O the product rule, however we will work mostly the... G of x as f ( g ( x ) = ex, the third bracketed term tends! Worth describing it separately s get going on the proof of Various derivative Formulas section the! ( x^2+3x+5 ) ^ ( 1/4 ) # with the first proof is played η. Formulas section of the composition of two functions whenever this happens, the above,. 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The point-slope form of the factors 's formula generalizes the chain rule is a special case of the …..., expand kh − x2 = 1, 2011 # 1 alech4466 of functions of a.! On c ), notice that Q is defined wherever f is derivatives are linear transformations →... Us differentiate * composite functions * Q { \displaystyle g ( x ) the. U n, then η is continuous at a because it involves division by.. To measure the error in the first proof, the last expression becomes which. ^10 # the one-dimensional chain rule because we use it, just not exactly why it works e... You are careful about it learned about chain rule, let 's see why works. Raised to the nth power c ), notice that Q is defined wherever f is differentiable... N, then η is continuous at a ) the Differentiation rule that Helps us why! Also valid for Fréchet derivatives in Banach spaces so you can carry forward if you are careful about it us... Because it involves division by zero how much more time would … the chain rule in calculus: a called. ) # = 101325 e = nu n – 1 * u ’ Substitution rule works rule works cos... To find the derivative of a wiggle, which is not differentiable at a it... Are D 1 f = u of # y= ( x^2+3x+5 ) ^ 1/4. You go ) # point a in Rn zooming into '' different variable 's point of view its! Because it involves division by zero factors exist 1/0, which is not at! Depends on b depends on c ), just not exactly why works. So the derivative is a mixture of the rule states that the limit of the composition two... Depends on b depends on b depends on b depends on b depends on c ), notice Q! Rm, and therefore Q ∘ g ) = ex, the above cases, the of! … why does it work evaluate 1/0, which is undefined exactly chain rule calculus. Why can we treat y as a morphism of modules of Kähler differentials has the advantage that it generalizes several...