![]() ![]() ![]() An informal proof is provided at the end of the section. Title: The Chain Rule Author: jsz Created Date: 1:47:41 PM. Calculus Stewart 6th Edition Section 2.5 The Chain Rule Appendixes A1, F Proofs of Theorems. The Theorem of Chain Rule: Let f be a real-valued function that is a composite of two functions g and h. It is used only to find the derivatives of the composite functions. Now that we have derived a special case of the chain rule, we state the general case and then apply it in a general form to other composite functions. In this example we must use the Product Rule before using the Chain Rule. This chain rule is also known as the outside-inside rule or the composite function rule or function of a function rule. As we determined above, this is the case for h(x)= \sin (x^3). We can take a more formal look at the derivative of h(x)= \sin (x^3) by setting up the limit that would give us the derivative at a specific value a in the domain of h(x)= \sin (x^3). So, you start with d/dx (x2+1)3 3 (x2+1)2 (2x) 6x (x2+1)2 (Chain Rule) Now, do that same type of process for the derivative of the second multiplied by the first factor. In addition, the change in x^3 forcing a change in \sin (x^3) suggests that the derivative of \sin (u) with respect to u, where u=x^3, is also part of the final derivative. Remember your product rule: derivative of the first factor times the second, plus derivative of the second factor times the first. First of all, a change in x forcing a change in x^3 suggests that somehow the derivative of x^3 is involved. This chain reaction gives us hints as to what is involved in computing the derivative of \sin (x^3). We can think of this event as a chain reaction: As x changes, x^3 changes, which leads to a change in \sin (x^3). Consequently, we want to know how \sin (x^3) changes as x changes. We can think of the derivative of this function with respect to x as the rate of change of \sin (x^3) relative to the change in x. To put this rule into context, let’s take a look at an example: h(x)= \sin (x^3). As the name suggests, antidifferentiation is the reverse process of differentiation. ![]() Instead, we use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. The antiderivative rules in calculus are basic rules that are used to find the antiderivatives of different combinations of functions. However, using all of those techniques to break down a function into simpler parts that we are able to differentiate can get cumbersome. We’ll solve this using three different approaches but we encourage you to become comfortable with the third approach as quickly as possible, because that’s the one you’ll use to compute derivatives quickly as the course progresses. Figure 2.When we have a function that is a composition of two or more functions, we could use all of the techniques we have already learned to differentiate it. The tangent line is sketched along with \(f\) in Figure 2.17. Thus the equation of the tangent line is \ ![]() To find \(f^\prime\),we need the Chain Rule. The tangent line goes through the point \((1,f(1)) \approx (1,0.54)\) with slope \(f^\prime(1)\). Find the equation of the line tangent to the graph of \(f\) at \(x=1\). This is called the Generalized Power Rule.Įxample 62: Using the Chain Rule to find a tangent line ![]()
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