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Chapter 3: Problem 1

Two equivalent forms of the Chain Rule for calculating the derivative of \(y=f(g(x))\) are presented in this section. State both forms.

Short Answer

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Question: State the two equivalent forms of the Chain Rule for calculating the derivative of a composite function \(y=f(g(x))\). Answer: 1. The first form of the Chain Rule is: $$(f(g(x)))' = f'(g(x)) \cdot g'(x)$$ 2. The second form of the Chain Rule using the Leibniz notation is: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

Step by step solution

01

Form 1: Chain Rule using the derivative of a composite function

The first form of the Chain Rule can be stated as follows: Let \(y = f(g(x))\), where \(f\) and \(g\) are functions with derivatives \(f'(g(x))\) and \(g'(x)\), respectively. The derivative of the composite function \(y\) with respect to \(x\) is given by: $$(f(g(x)))' = f'(g(x)) \cdot g'(x)$$ This form of the Chain Rule states that the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function, and the derivative of the inner function.
02

Form 2: Chain Rule using the Leibniz notation

The second form of the Chain Rule can be stated using the Leibniz notation. Let \(y = f(u)\) and \(u = g(x)\), where \(f\) and \(g\) are functions with derivatives \(\frac{dy}{du}\) and \(\frac{du}{dx}\), respectively. We want to find the derivative \(\frac{dy}{dx}\) of the composite function \(y\). We can write the Chain Rule in terms of these derivatives as: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$ This form of the Chain Rule states that the derivative of a composite function with respect to \(x\) is the product of the derivative of the outer function with respect to its inner function \(u\) and the derivative of the inner function with respect to \(x\).

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Most popular questions from this chapter

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