Skill: Differentiate power functions

Explanation and Free Practice Resources

For a power function $f(x)=x^n$ , differentiation gives $f'(x)=nx^{n-1}$ , so the derivative represents the instantaneous rate of change or tangent slope and can be used with constant multiples and sums of power terms. The rule applies to standard integer powers, including zero and negative powers where the function is defined, with simple rational powers when included in the introductory domain; generalized real or complex exponents are not included.

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Detailed Explanation: Differentiate power functions

For a power function

f(x)=x^n,

use the power rule:

\frac{d}{dx}\left(x^n\right)=nx^{n-1}.

This means: bring the exponent down as a multiplier, then subtract 1 from the exponent. For sums and constant multiples, differentiate each term separately.

Example

Find the derivative of

f(x)=4x^3-7x^2+5x-9.

Differentiate one term at a time:

For $4x^3$ :

\frac{d}{dx}(4x^3)=4(3x^2)=12x^2.

For $-7x^2$ :

\frac{d}{dx}(-7x^2)=-7(2x)=-14x.

For $5x$ :

Since $x=x^1$ ,

\frac{d}{dx}(5x)=5(1x^0)=5.

For the constant $-9$ :

\frac{d}{dx}(-9)=0.

Putting the results together,

\boxed{f'(x)=12x^2-14x+5}.

The derivative $f'(x)$ gives the instantaneous rate of change of $f$ at any value of $x$ . For example, the tangent slope at $x=1$ is

f'(1)=12(1)^2-14(1)+5=3.

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