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Chapter 6: Problem 25

Let \(f\) be a function for which \(f^{\prime}(x)=x^{2}+1\) If \(y=f\left(\sin \left(x^{3}\right)\right)\), find \(\frac{d y}{d x}\).

Short Answer

Expert verified
\(\frac{dy}{dx} = 3x^{2}\cos(x^{3})(\sin^{2}(x^{3}) + 1)\

Step by step solution

01

Identify the Inner and Outer Functions

In this exercise, \(y = f(\sin(x^{3}))\), the inner function is \(u = \sin(x^{3})\) and the outer function is \(f(u)\).
02

Apply the Chain Rule

The chain rule states that \(\frac{dy}{dx} = f'(u) \cdot \frac{du}{dx}\), where \(f'(u)\) represents the derivative of the outer function and \(\frac{du}{dx}\) is the derivative of the inner function. According to the derivative function given in the problem, \(f'(u) = u^{2} + 1\), and \(\frac{du}{dx} = \frac{d}{dx}\sin(x^{3}) = 3x^{2}\cos(x^{3})\) (This is obtained by applying the chain rule once more for \(\sin(x^{3})\)).
03

Substitution and Simplification

Now, substitute the inner function \(u = \sin(x^{3})\) and its derivative into the chain rule equation: \(\frac{dy}{dx} = (\sin(x^{3})^{2} + 1) \cdot 3x^{2}\cos(x^{3})\). Then, further simplify it as \(\frac{dy}{dx} = 3x^{2}\cos(x^{3})(\sin^{2}(x^{3}) + 1)\).

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