Problem 39 - College Algebra and Calculus: An Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ h(x)=\left(4-x^{3}\right)^{-4 / 3} $$

Step-by-Step Solution

Answer

The derivative of the function \( h(x) = (4-x^{3})^{-4 / 3} \) is \( dh/dx = 4x^{2}(4 - x^{3})^{-7/3} \) .

Step 1: Identify the base function and its power

The function is given as \( h(x) = (4-x^{3})^{-4 / 3} \). Here, \( u = 4-x^3 \) is the base function and it's raised to the power of -4/3.

Step 2: Apply the General Power Rule

The General Power Rule states that the derivative of \( u^n \) with respect to \( x \) is \( n \cdot u^{n-1} \cdot (du/dx) \) . Apply this rule to the identified function. \( dh/dx = -4/3 \cdot (4 - x^{3})^{-7/3} \cdot (du/dx) \). We still need to find \( du/dx \).

Step 3: Find the derivative of \( u \)

We find the derivative of the base function \( u \), which is the inner part of our given function. \( du/dx = -3x^{2} \).

Step 4: Substitute \( du/dx \) into the derivative formula

After finding \( du/dx \), we substitute it into the previous formula resulting in the derivative of the function \( h(x) \). \( dh/dx = -4/3 \cdot (4 - x^{3})^{-7/3} \cdot (-3x^{2}) = 4x^{2}(4 - x^{3})^{-7/3} \) .