The Power Rule and Basic Differentiation Rules

By the end of this lesson, you will be able to apply the Power Rule, Constant Rule, Sum Rule, and Constant Multiple Rule to differentiate any polynomial or simple algebraic function without the limit definition.

The Power Rule

For any real number $n$:

$$\frac{d}{dx}[x^n] = nx^{n-1}$$

Examples:

$\dfrac{d}{dx}[x^5] = 5x^4$

$\dfrac{d}{dx}[x^{-2}] = -2x^{-3} = -\dfrac{2}{x^3}$

$\dfrac{d}{dx}[\sqrt{x}] = \dfrac{d}{dx}[x^{1/2}] = \dfrac{1}{2}x^{-1/2} = \dfrac{1}{2\sqrt{x}}$

$\dfrac{d}{dx}[x] = 1$ (power rule with $n=1$)

$\dfrac{d}{dx}[1] = \dfrac{d}{dx}[x^0] = 0$ (consistent with constant rule)

Basic Rules

Constant Rule: $\dfrac{d}{dx}[c] = 0$

Constant Multiple Rule: $\dfrac{d}{dx}[cf(x)] = cf'(x)$

Sum/Difference Rule: $\dfrac{d}{dx}[f \pm g] = f' \pm g'$

These rules allow differentiation of any linear combination of power functions — i.e., any polynomial.

Example 1 — Polynomial

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

$f'(x) = 20x^4 - 9x^2 + 7$

Example 2 — Rational and Radical

$g(x) = \dfrac{3}{x^2} + 5\sqrt[3]{x}$

$= 3x^{-2} + 5x^{1/3}$

$g'(x) = -6x^{-3} + \dfrac{5}{3}x^{-2/3} = -\dfrac{6}{x^3} + \dfrac{5}{3x^{2/3}}$

Example 3 — Finding Slope and Tangent

$h(x) = x^4 - 6x^2 + 9$. Find the slope at $x = 2$ and the tangent line.

$h'(x) = 4x^3 - 12x$

$h'(2) = 32 - 24 = 8$

$h(2) = 16 - 24 + 9 = 1$

Tangent: $y - 1 = 8(x-2) \implies y = 8x - 15$

Example 4 — Where is the Slope Zero?

$f(x) = x^3 - 3x + 2$. Find $x$ where $f'(x) = 0$.

$f'(x) = 3x^2 - 3 = 3(x^2-1) = 3(x-1)(x+1) = 0$

$x = 1$ and $x = -1$ — these are critical points (potential local extrema).

Higher-Order Derivatives

$f''(x) = \dfrac{d}{dx}[f'(x)]$, $f'''(x) = \dfrac{d}{dx}[f''(x)]$, etc.

For $f(x) = x^4$: $f'(x) = 4x^3$, $f''(x) = 12x^2$, $f'''(x) = 24x$, $f^{(4)}(x) = 24$, $f^{(5)}(x) = 0$.

Common Mistakes