Mathfolis

Power Rule

Unit 2 · Differentiation — Rules

The Power Rule

The Power Rule is the most fundamental differentiation shortcut. Combined with the sum/difference and constant multiple rules, it lets you differentiate any polynomial instantly.

Power Rule

For any real n
ddx[xn]=nxn−1\frac{d}{dx}[x^n] = nx^{n-1}

Supporting Rules

Constant
ddx[c]=0\frac{d}{dx}[c] = 0
Constant Multiple
ddx[c⋅f(x)]=c⋅f′(x)\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)
Sum / Difference
ddx[f±g]=f′±g′\frac{d}{dx}[f \pm g] = f' \pm g'

Tangent Line at x = a

y−f(a)=f′(a)(x−a)y - f(a) = f'(a)(x - a)
AP Tip: Always rewrite radicals and fractions as powers before differentiating: √x = x^(1/2), 1/x³ = x^(−3).
Type 1

Basic Power Rule — Polynomials

Apply the power rule term by term to differentiate polynomial and rational exponent expressions.

Example 1

Find the derivative of f(x) = 4x⁵ − 3x² + 7x − 1.

f(x)=4x5−3x2+7x−1f(x) = 4x^5 - 3x^2 + 7x - 1
Example 2

Find the derivative. Rewrite using power notation first.

ddx ⁣[1x3+x]\frac{d}{dx}\!\left[\frac{1}{x^3} + \sqrt{x}\right]

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Type 2

Tangent Line Equation

Find the slope using the derivative, then use point-slope form to write the tangent line equation.

Example 3

Find the equation of the tangent line at x = 2.

f(x)=x3−2x+1f(x) = x^3 - 2x + 1

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Type 3

Differentiability vs. Continuity

Differentiability implies continuity, but not vice versa. Corners, cusps, and vertical tangents make a continuous function non-differentiable.

Example 4

Is f(x) = |x − 2| differentiable at x = 2? Is it continuous?

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