Power Rule

Core Definition

For any real number $n \in \mathbb{R}$ , the derivative of the power function $f(x) = x^n$ with respect to $x$ is given by:

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

Combined with the Constant Rule (

\frac{d}{dx}[c] = 0

), Constant Multiple Rule (

\frac{d}{dx}[c \cdot f(x)] = c f'(x)

), and Sum/Difference Rules (

\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)

), the Power Rule enables rapid differentiation of any polynomial, radical, or rational power term without computing limit difference quotients from scratch.

Concept Explanation

1. The Power Rule Family of Basic Rules

Constant Rule: The graph of $f(x) = c$ is a horizontal line with zero slope everywhere: $\frac{d}{dx}[c] = 0$
Linear Identity Rule: The graph of $f(x) = x = x^1$ is a line with constant slope $1$ : $\frac{d}{dx}[x] = 1 \cdot x^0 = 1$
Constant Multiple Rule: Multiplicative constants factor outside the derivative operator: $\frac{d}{dx}[c \cdot x^n] = c \cdot (n x^{n-1})$
Sum & Difference Rule: Differentiate term-by-term across addition and subtraction: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$

2. The Algebraic Pre-Processing Protocol

Many functions do not immediately appear in the base power form $x^n$ . Before applying differentiation rules, algebraically rewrite expressions into explicit power form:

Rational Exponent Conversion: Transform radical roots into fractional powers: $\sqrt[n]{x^m} = x^{\frac{m}{n}} \quad \text{e.g., } \sqrt{x} = x^{\frac{1}{2}}, \quad \sqrt[3]{x^2} = x^{\frac{2}{3}}$
Negative Exponent Conversion: Move variables out of denominators into the numerator: $\frac{1}{x^n} = x^{-n} \quad \text{e.g., } \frac{5}{x^3} = 5x^{-3}, \quad \frac{2}{\sqrt{x}} = 2x^{-\frac{1}{2}}$
Polynomial Expansion / Monomial Splitting: Expand products or split fractions with a single denominator term: $\frac{3x^4 - 2x + 1}{x^2} = \frac{3x^4}{x^2} - \frac{2x}{x^2} + \frac{1}{x^2} = 3x^2 - 2x^{-1} + x^{-2}$

3. The Algebraic Post-Processing Protocol

After taking the derivative, terms with negative or fractional exponents should be factored and simplified for subsequent evaluation (e.g., finding critical points or higher derivatives):

Factoring Out Lowest Powers: When terms have fractional exponents, factor out the smallest (most negative) exponent. Factoring out $x^k$ subtracts $k$ from the remaining exponents: $3x^{-\frac{1}{2}} + 6x^{\frac{1}{2}} = 3x^{-\frac{1}{2}}\left(1 + 2x^{\frac{1}{2} - (-\frac{1}{2})}\right) = 3x^{-\frac{1}{2}}(1 + 2x^1) = \frac{3(1+2x)}{\sqrt{x}}$

Engineering Applications

Civil Engineering (Structural Beam Shear): The internal shear force $V(x)$ in a bridge girder is the spatial derivative of the bending moment polynomial $M(x)$ : $V(x) = \frac{dM}{dx}$ .
Aerospace Engineering (Atmospheric Density & Drag): Atmospheric pressure and density profiles modeled by polytropic power laws $\rho(z) = \rho_0 \left(1 - \frac{\alpha z}{T_0}\right)^n$ are differentiated with respect to altitude $z$ to compute drag variation during ascent.
Electrical Engineering (Capacitive Energy Dissipation): The instantaneous power in a circuit element with energy $E(t) = \frac{1}{2} C [v(t)]^2$ is evaluated using power rule derivatives.

Example Problems

Example 1: Differentiating Radicals and Rational Expressions

Find the derivative of $f(x) = 5\sqrt[3]{x^2} - \frac{4}{x^3} + \frac{7}{\sqrt{x}} + 9$ .

Solution

Pre-Process the expression into power form: $f(x) = 5x^{\frac{2}{3}} - 4x^{-3} + 7x^{-\frac{1}{2}} + 9$
Apply the Power Rule and Constant Multiple Rule term-by-term: $f'(x) = 5\left(\frac{2}{3}x^{\frac{2}{3} - 1}\right) - 4\left(-3x^{-3-1}\right) + 7\left(-\frac{1}{2}x^{-\frac{1}{2}-1}\right) + 0$ $f'(x) = \frac{10}{3}x^{-\frac{1}{3}} + 12x^{-4} - \frac{7}{2}x^{-\frac{3}{2}}$
Post-Process back into standard radical and fractional notation: $f'(x) = \frac{10}{3\sqrt[3]{x}} + \frac{12}{x^4} - \frac{7}{2\sqrt{x^3}}$

Example 2: Monomial Denominator Division

Find the derivative of $g(t) = \frac{4t^5 - 2t^2 + 7\sqrt{t}}{t^2}$ .