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Calculus basics

Power rule for derivatives

To differentiate a power of x, bring the power down as a multiplier and lower it by one.

ddx(axn)=naxn−1
Constantddx(c)=0
Sum ruleddx(f+g)=f′+g′

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

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  • At x = 4: f(x) = 48, slope f′(x) = 24 (coral tangent). Dashed: the derivative f′.

What the symbols mean

m
the slope (gradient) of the curve at the chosen x: the value of f′(x)
a
the coefficient in f(x)=axn
n
the power (any real number)
x
the point where you want the slope

When to use it

Use it on every polynomial term, and on roots and reciprocals once they are written as powers: x=x1/2, 1/x2=x−2. With the sum rule it differentiates any polynomial term by term. Evaluating the derivative at a point gives the slope of the curve there.

Worked example

Find the slope of f(x) = 3x² at x = 4 (a = 3, n = 2).

  1. Start from the formula.

    m=n⁢a⁢xn−1

  2. Substitute a = 3, n = 2, x = 4 and work it out.

    m=2·3·42−1
    =6·41
    =6·4
    =24

  3. In general, the derivative of 3⁢x2 is f′(x)=6⁢x; the number above is its value at x=4, the gradient of the curve there.

Answer f′(4) = 24: the curve rises 24 units per unit of x there

Why it works

For x2, (x+h)2−x2h=2x+h, which tends to 2x as h shrinks. In general, expanding (x+h)n with the binomial theorem leaves nxn−1 plus terms that vanish as h→0.

Common mistakes

  • Lowering the power but forgetting to multiply by it: the derivative of x3 is 3x2, not x2.
  • Differentiating x or 1/x without rewriting them as powers first.
  • Thinking the derivative of a constant is the constant; it is 0.
  • Applying it to 2x (variable exponent); that needs the exponential rule.