Problem 35 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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, \(\frac{du}{dx} = 18x\).

Step 3: Differentiate Outer Function

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

Step 4: Apply General Power Rule

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