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Differentiation rules for polynomials

The derivative $f'(x)$ is the instantaneous rate of change, or the slope of the tangent at the point. For polynomials you differentiate term by term: the exponent moves down as a factor and is lowered by one. A constant term has derivative zero, and a constant factor stays.

Try it yourself

The derivative as a slope

(x^n)' = n\,x^{n-1}

power rule

(k\cdot f)' = k\cdot f'

constant factor

(c)' = 0

constant term

Symbols

$n$	exponent
$k,\ c$	constants

Example

$f(x) = 3x^3 - 2x + 5$ gives $f'(x) = 9x^2 - 2$ , and $f'(2) = 9\cdot 4 - 2 = 34$ .

Split into terms and differentiate each term separately.

Practise differentiation and rate of change for free →

← Average rate of change · Equation of the tangent →

Part of Mathematics 1T: Differentiation and rate of change.