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Finding the Derivative of e^x

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After you read this lesson, you will know how to quickly find the derivative of e^x. You'll learn that your answer is actually already written out for you and all you have to do is to write down as you see it.

The Steps

There is only one step you need to take to find the derivative of e^x, and this is it.

  • Step 1: The derivative of e^x is e^x.

That's right. All you have to do is to write down e^x again as the derivative of e^x is itself. Why is this so?

There are various ways to prove this. The simplest is knowing that the graph of e^x is a very unique graph. If you calculate the slope at each point of the graph, you'll find that the slope at that point will equal the function's value. Since the slope is the same as the function at every point on the graph, then the derivative of this function will be itself, since the derivative gives you the slope of the function.


e^x Graph
e^x graph


Another proof involves using the natural log and substitution.

If you let e^x = y, then you can take the natural log of both sides.


e^x


Next, you take the derivative on both sides with respect to x.


e^x


Finally, multiplying both sides by y and then substituting the e^x back in for y, you get the proof you need that the derivative of e^x is equal to itself.


e^x


The Solution

We have seen that the derivative of e^x is e^x.

Let's take a look at how you can use this information in a problem.


The Problem
e to the x


To find the derivative of this problem, you follow all the rules for taking derivatives. So if you see a constant like the 2, you can move that to the other side of the derivative sign.


Solving The Problem
e to the x


Now you can go ahead and evaluate the derivative of e^x. You remember that this derivative is actually itself, so your answer is 2e^x. And you are done.

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