The Power Rule & Basic Rules

You’ll be able to

Differentiate power functions with the power rule, including negative and fractional exponents
Apply the constant-multiple and sum/difference rules
Rewrite roots and reciprocals as powers before differentiating

The power rule

The power rule is the workhorse of differentiation: for any real exponent n, d/dx[xⁿ] = n·xⁿ⁻¹. Bring the exponent down as a coefficient and drop the exponent by one. It works for any constant exponent — positive, negative, or fractional — which is why rewriting a root or a reciprocal as a power is the key first step. The derivative of a constant is 0, since a constant is x⁰ and has zero slope everywhere.

Constant multiples and sums

Two rules let you break a function into manageable pieces. The constant-multiple rule: d/dx[c·f(x)] = c·f′(x) — a constant factor rides along untouched. The sum/difference rule: d/dx[f ± g] = f′ ± g′ — you may differentiate term by term. Together they mean any polynomial is differentiated one term at a time, which makes polynomials the easiest functions to handle.

Rewrite before you differentiate

The power rule only sees exponents, so translate everything into exponent form first. A square root becomes a ½ power: √x = x^(1/2). A reciprocal becomes a negative power: 1/x³ = x⁻³. Once rewritten, apply the power rule mechanically. Forgetting to rewrite — and instead guessing — is the most common source of errors on these problems.

Power rule

d/dx[xⁿ] = n·xⁿ⁻¹

Works for every constant n. Special cases: d/dx[c] = 0 and d/dx[x] = 1.

Linearity of the derivative

d/dx[c·f(x)] = c·f′(x) and d/dx[f(x) ± g(x)] = f′(x) ± g′(x)

Constants pass through; sums and differences split apart. Differentiate a polynomial term by term.

Worked example

Differentiate f(x) = 4x³ − 2/x + 6√x.

1.Rewrite each term as a power: 4x³ − 2x⁻¹ + 6x^(1/2).
2.Power rule on 4x³: 4·3·x² = 12x².
3.Power rule on −2x⁻¹: −2·(−1)·x⁻² = 2x⁻² = 2/x².
4.Power rule on 6x^(1/2): 6·(1/2)·x^(−1/2) = 3x^(−1/2) = 3/√x.
5.Combine the terms.

Answer: f′(x) = 12x² + 2/x² + 3/√x.

Tip

Always convert roots and fractions to exponents before differentiating. √x → x^(1/2) and 1/x² → x⁻² turn intimidating terms into one-step power-rule applications, and they keep negative-exponent bookkeeping straight.

Checkpoint

What is the derivative of f(x) = 7x⁴ − 3x + 9?

Checkpoint

What is d/dx[1/x²]?

On the exam

Watch the sign and the new exponent carefully with negative powers. d/dx[x⁻²] = −2x⁻³ — the coefficient is negative and the exponent becomes more negative. Rushing this step is a frequent multiple-choice trap.

Answer the 2 checkpoints as you read.