Problem 38 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Outer and Inner Functions

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

Step 2: 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.

Step 3: 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: \(f'(x)= -1/2 * (25+x^{2})^{-3 / 2} * 2x\)

Step 4: 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.