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The power rule

The derivative tells how steep the graph is, that is how fast the function value changes. For a power you bring the exponent down as a factor and reduce the exponent by 1. A constant in front of x stays. A constant on its own has derivative zero, because a horizontal line does not rise. Calculate the function value and the derivative separately when the task asks for both.

(ax^n)' = anx^{n-1}

the power rule

(c)' = 0

a constant

Symbols

$a$	coefficient
$n$	exponent

Example

$f(x) = 2x^3$ . Then $f(2) = 2 \cdot 2^3 = 16$ and $f'(x) = 6x^2$ , so $f'(2) = 6 \cdot 2^2 = 24$ .

The exponent decreases by 1. If you forget that, the derivative is wrong.

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Part of Mathematics S1: Differentiation and optimisation.