Differentiation Rules

The Power Rule

Find the derivative of a power function using the power rule.

power-rule The power rule states that if f(x) = xⁿ, then f'(x) = n·xⁿ⁻¹. Bring down the exponent as a coefficient, then reduce the exponent by one.

Example

Bring down the exponent, then reduce the exponent by one. This shortcut abbreviates the limit definition of the derivative; the book applies the rule but does not prove it from that limit.

highlighted = computed this step

Step 1 — Set up

Start with x to the 5th power.

f(x)=x5f(x)= x^{5}

Step 2 — Power rule

Use the power rule: bring the exponent down.

ddxxn=nxn−1\frac{d}{dx}x^{n}=n x^{n- 1 }

Step 3 — Apply the rule

Bring down 5 and subtract 1 to get exponent 4.

5x4\hl{5} x^{ \hl{4} }

Step 4 — Result

State the derivative.

f′(x)=5x4f'(x)= \hlmath{5x^{4}}