Power Rule for Derivatives: Step-by-Step Guide
The power rule states that d/dx[x^n] = n*x^(n-1) for any real exponent n. Combined with the constant multiple, sum, and difference rules, it lets you differentiate every polynomial and any function expressible as a sum of power terms.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Rewrite each term as a power of x. Convert radicals and fractions to exponent form: sqrt(x) = x^(1/2), 1/x^k = x^(-k), 1/sqrt(x) = x^(-1/2). This puts every term in the form c*x^n.\[\sqrt[m]{x^k} = x^{k/m}, \;\; \frac{1}{x^k} = x^{-k}\]
- Apply the power rule term by term. For each term c*x^n, the derivative is c*n*x^(n-1). Multiply the coefficient by the exponent and reduce the exponent by 1.\[\frac{d}{dx}[c\,x^n] = c\,n\,x^{n-1}\]
- Differentiate constants to zero. Any constant term has derivative 0. The constant multiple rule lets you pull constants outside of the derivative.\[\frac{d}{dx}[c] = 0\]
- Combine and simplify. Sum the term derivatives. Optionally rewrite negative or fractional exponents as radicals/fractions if a 'cleaner' form is desired.\[\frac{d}{dx}\left[\sum c_i x^{n_i}\right] = \sum c_i\,n_i\,x^{n_i - 1}\]
Worked example
Find d/dx of f(x) = 4x^3 - 2*sqrt(x) + 5/x^2 - 7.
Step 1: Rewrite each term as a power of x: f(x) = 4x^3 - 2x^(1/2) + 5x^(-2) - 7.
Step 2: Apply the power rule to each term. d/dx[4x^3] = 12x^2.
Step 3: d/dx[-2x^(1/2)] = -2 * (1/2) * x^(-1/2) = -x^(-1/2) = -1/sqrt(x).
Step 4: d/dx[5x^(-2)] = 5 * (-2) * x^(-3) = -10x^(-3) = -10/x^3.
Step 5: d/dx[-7] = 0 (derivative of a constant).
Step 6: Combine: f'(x) = 12x^2 - 1/sqrt(x) - 10/x^3.
Verification: At x = 1, f'(1) = 12 - 1 - 10 = 1. Numerical check: (f(1.001) - f(1))/0.001 approximately 1.001, confirming the answer.
Tips and common mistakes
- The power rule works for ALL real exponents — positive, negative, fractional, and irrational.
- Always rewrite radicals and reciprocals as exponents BEFORE differentiating; never try to differentiate sqrt(x) or 1/x as-is.
- Multiply the coefficient by the original exponent, then subtract 1 from the exponent.
- The derivative of a constant is 0; the derivative of x by itself (x^1) is 1.
- Sum/difference rule: differentiate each term independently and add/subtract the results.
- Constant-multiple rule: d/dx[c f(x)] = c f'(x); you can pull constants outside.
- Be careful with negative exponents: subtracting 1 makes them more negative (e.g., -2 becomes -3).
- Power rule does NOT apply to a^x with constant base — for that, use the exponential derivative formula.