Find the derivative of [f(x)]n where f(x) is differentiable and n is an integer.

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Lesson Parent: 
The Chain Rule
Problem: 

Find the derivative of [f(x)]n where f(x) is differentiable and n is an integer.

Answer: 

ddx[[f(x)]n]=n[f(x)]n−1⋅dfdx

SAMPLE SOLUTION

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Solution: 

Let's define <Sign in to see all the formulas>. The function composition <Sign in to see all the formulas>. We know that u(x) is differentiable and by assumption that f(x) is differentiable. We can therefore use the Chain Rule. Let's follow our steps outlined in our Chain Rule lesson:

Step Example
Determine <Sign in to see all the formulas>. We know f(x) is differentiable but we don't know its formula.

<Sign in to see all the formulas>

Place f in for every value of x in <Sign in to see all the formulas>. Call this <Sign in to see all the formulas>.

<Sign in to see all the formulas>

Place the function f(x) into every value of f in <Sign in to see all the formulas>. We still call this <Sign in to see all the formulas>.

<Sign in to see all the formulas>

Multiply <Sign in to see all the formulas> times <Sign in to see all the formulas>. This is <Sign in to see all the formulas>.

<Sign in to see all the formulas>