2.5 Applying the Power Rule

The Power Rule

For any real number n, if f(x) = x^n, then f'(x) = n·x^(n-1). This works for positive, negative, and fractional exponents. Bring the exponent down as a coefficient, then subtract 1 from the exponent. It's the fastest way to differentiate power functions like x^5, x^(1/2), or x^(-3).

Rewriting before differentiating

Radicals and fractions must be rewritten as powers of x before applying the rule. √x = x^(1/2), 1/x^3 = x^(-3). Constants multiply through: d/dx[c·x^n] = c·n·x^(n-1). Always simplify the expression first — trying to differentiate √x or 1/x directly without rewriting leads to mistakes.

Combining with sum/difference rule

Polynomials are differentiated term by term using the sum and difference rule alongside the power rule. The derivative of a constant is 0 (since x^0 has n=0). For f(x) = 3x^4 - 2x + 7, differentiate each term: f'(x) = 12x^3 - 2 + 0 = 12x^3 - 2.

Key terms

Power Rule

d/dx[x^n] = n·x^(n-1), for any real number n.

Derivative of a constant

0 — since a constant can be written as c·x^0, and the power rule gives 0.

How to differentiate √x

Rewrite as x^(1/2), then apply the power rule: (1/2)x^(-1/2).

How to differentiate 1/x^n

Rewrite as x^(-n), then apply the power rule: -n·x^(-n-1).

Sum/Difference Rule with Power Rule

Differentiate polynomials term by term; each term's derivative is found separately using the power rule.

Try an AP-style question

d/dx (1/x³) = ?

1. −3x⁻⁴
2. 3x⁻⁴
3. 1/(3x²)
4. −3x⁻²