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In this lesson, we show how to take the derivative of an exponential where the argument of the exponential is a specific function of x. After explaining the steps, we verify the result using graphical examples.
The Derivative
Although there are no parentheses in the expression, e4x, we can think of this as a function of a function. The exponential function has a function of x in its argument. With parentheses we would write e(4x).
Another perspective identifies an outer function and an inner function. The outer function is e(x) and the inner function is 4x. The derivative of e(x) is e(x). The derivative of 4x is 4.
We find the derivative of e4x using two steps:
Step 1: Use the Chain Rule.
The chain rule says when we have an outer function and an inner function, we get the derivative by multiplying the derivative of the outer function by the derivative of the inner function.
Step 2: Simplify.
Usually, simplifying means to rearrange an expression into a commonly acceptable form.
Let’s work through these two steps:
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On the first line, we see the derivative of the outer function, e4x, multiplied by the derivative of the inner function. On the second line, the inner function, 4x, differentiates to 4. On the third line, the expression is simplified by moving the 4 in front so it acts as a coefficient for e4x.
The Final Result
So the derivative of e4x is given by:
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Graphical Examples
Let’s say we wanted to compare e4x with its derivative, 4e4x. One way to do this is to plot both functions on the same graph.
The function and its derivative
plot_of_function_and_its_derivative
 
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