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.

differentiating e

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


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


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 


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.

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