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Differentiating e to the kx

The function y=e^{kx}, where k is a constant, can be found using the chain rule. It is useful to remember the following:

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}


Example 1Differentiate y=2e^{4x}.

\frac{dy}{dx}=4\times 2e^{4x}=8e^{4x}

Example 2Given that

h(x)=\frac{1+20e^{5x}}{4e^{2x}}.

Find h'(x).

h(x) can be written as

h(x)=\frac{1}{4e^{2x}}+\frac{20e^{5x}}{4e^{2x}}=\frac{1}{4}e^{-2x}+5e^{3x}.

We can differentiate more simply when h is written in this format:

h

Example 3 After time t seconds, the temperature, T degrees, of a heated metal ball that is dropped into a liquid is given by

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

Find the temperature of the ball at the instant it is dropped into the liquid. Find the rate at which the ball is cool after 10 seconds.

Substituting t=0 into T gives the intial temperature as 377 degrees. The rate of cooling is given by the derivative of T:

T

At time t=10 seconds, the rate of cooling is

T

to 2 decimal places. This means that the ball is cooling by 12.88 degrees per second but this is only true at the instant 10 seconds after the ball is dropped into the liquid.