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Chapter 7: Problem 39

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

Short Answer

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

Step by step solution

01

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.
02

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 \( du/dx \).
03

Find the derivative of \( u \)

We find the derivative of the base function \( u \), which is the inner part of our given function. \( du/dx = -3x^{2} \).
04

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 \( h(x) \). \( dh/dx = -4/3 \cdot (4 - x^{3})^{-7/3} \cdot (-3x^{2}) = 4x^{2}(4 - x^{3})^{-7/3} \) .

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Most popular questions from this chapter

Use the given information to find \(f^{\prime}(2)\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=\frac{g(x)}{h(x)} $$

Use the General Power Rule to find the derivative of the function. $$ f(x)=\left(4 x-x^{2}\right)^{3} $$

Use a graphing utility to graph \(f\) and \(f^{\prime}\) on the interval \([-2,2]\). $$ f(x)=x(x+1) $$

Use a graphing utility to graph \(f\) and \(f^{\prime}\) on the interval \([-2,2]\). $$ f(x)=x^{2}(x+1)(x-1) $$

Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule $$ f(x)=\sqrt[3]{x^{2}} $$
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