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Chapter 7: Problem 35

Use the General Power Rule to find the derivative of the function. $$ y=\sqrt[3]{9 x^{2}+4} $$

Short Answer

Expert verified
The derivative of the function \(y = \sqrt[3]{9x^2 + 4}\) is \(y' = 6x (9x^2 + 4)^{-2/3}\)

Step by step solution

01

Identify the Inner and Outer Functions

The inner function \(u\) is \(9x^2 + 4\). The outer function \(y\) is \(u^{1/3}\). The General Power Rule will be used to find the derivative of both these functions.
02

Differentiate Inner Function

The derivative of \(u\) with respect to \(x\), denoted as \(u'\) or \(\frac{du}{dx}\), can be obtained by applying the power rule for each term individually: So, \(\frac{du}{dx} = 18x\).
03

Differentiate Outer Function

Differentiating the outer function \(y = u^{1/3}\) with respect to \(u\) gives \(\frac{dy}{du} = \frac{1}{3} u^{-2/3}\).
04

Apply General Power Rule

We can now combine steps 2 and 3, by multiplying \(\frac{dy}{du}\) and \(\frac{du}{dx}\) to get \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{1}{3} u^{-2/3} \cdot 18x = 6x (9x^2 + 4)^{-2/3}\).

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Most popular questions from this chapter

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