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Chapter 7: Problem 38

Use the General Power Rule to find the derivative of the function. $$ f(x)=\left(25+x^{2}\right)^{-1 / 2} $$

Short Answer

Expert verified
The derivative of the given function is \(f'(x)= -x(25+x^{2})^{-3 / 2}\).

Step by step solution

01

Identify the Outer and Inner Functions

The outer function here can be considered as \(u^{-1/2}\) while the inner function is \(u = 25+x^{2}\)
02

Apply the General Power Rule

According to the General Power Rule also known as Chain Rule, the derivative of the function \(y(u) = u^{-1/2}\) is \(y'(u) = -1/2 * u^{-3/2}\). Apply this to our outer function while leaving the inner function as is. Doing this grants: \(f'(x)= -1/2 * (25+x^{2})^{-3 / 2}\). However, we must remember to multiply this by the derivative of the inner function which brings us to the next step.
03

Derive the Inner Function

The derivative of the inner function \(u = 25+x^{2}\) is \(u' = 2x\). Multiply this derivative by the one we obtained in Step 2. Doing so grants: \(f'(x)= -1/2 * (25+x^{2})^{-3 / 2} * 2x\)
04

Simplify the Result

The function will simplify to: \(f'(x)= -x(25+x^{2})^{-3 / 2}\). This now is the derivative of the original function.

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Most popular questions from this chapter

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