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The derivative of the exponential

The exponential function with base $e$ is its own derivative. If there is a coefficient in the exponent, it appears as a factor because of the chain rule. You therefore differentiate the exponent and multiply it onto the original function. That makes $e^{kx}$ easy to differentiate several times. Insert a number at the end if the task asks for a particular slope.

Try it yourself

The derivative as a slope

(e^x)' = e^x

its own derivative

(e^{kx})' = ke^{kx}

with a coefficient

Symbols

$k$	the coefficient in the exponent
$e$	the base

Example

$(e^{3x})' = 3e^{3x}$ , which is $3$ when $x = 0$ . $(e^{2x})'$ at $x = 1$ is $2e^{2} \approx 14.78$ .

The coefficient in the exponent must be multiplied on, not left behind as an exponent on its own.

Practise differentiation rules in practice for free →

← The definition of the derivative · The derivative of ln →

Part of Mathematics R1: Differentiation rules in practice.