# Differentiating e to the kx

Differentiating e to the x does not change the function, i.e. y and dy/dx are the same expression. The graph below shows the graph of $y=e^{x}$ where $e$, sometimes known as Euler’s number, is given by $e=2.718281828459$… See more on this type of graph. The number $e$ is special because everywhere on this graph, the gradient is the same as the $y$-coordinate.

The derivative of $y=e^{kx}$, where $k$ is a constant, is $\frac{dy}{dx}=ke^{kx}$, i.e.:

$y=e^{kx},\hspace{10pt}\frac{dy}{dx}=ke^{kx}$

For $k=1$, this says that at each point on the graph of $y$, the gradient matches that of the $y$ coordinate:

$y=e^{x},\hspace{10pt}\frac{dy}{dx}=e^{x}$

Be careful when differentiating multiples of these functions. The power and multiple must be multiples together. See the Examples.

## Examples of Differentiating e to the kx

Differentiate $y=2e^{4x}$.

Given that $h(x)=\frac{1+20e^{5x}}{4e^{2x}}$,
find $h'(x)$.

A scientist drops a heated metal ball into a cooler liquid. After time $t$ seconds, the temperature, $T^\circ$, of the ball is

$T(t)=350e^{-0.1t}+27$.

Find the temperature of the ball at the instant the scientist drops the ball into the liquid. Find the rate at which the ball is cooling after 10 seconds.