Chain Rule: The General Exponential Rule - Problem 1
A lot of really simple exponential functions fall under the category of general exponential function, which I call functions of the form e to the g(x). I'm going to deal with a bunch of examples that are e to the, and then constant times x, because they’re all differentiated the same way. We use the general exponential rule, but it will be a really simple version of it. So, when you’re differentiating something like e to the -2x, the derivative is e to the -2x, times the derivative of -2x.
The derivative of -2x is -2. So you’re just going to get e to the -2x times -2, or -2e to the -2x. And this is something you’ll see a lot. So let’s take a look at three more examples that are kind of like this. The derivative of e to the 0.05x. Same thing. We have like a little m times x here and the derivative is going to be e to the 0.05x times the derivative of 0.05x which is 0.05.
And then here just in general, if you just have a value mx, it’s going to be e to the mx times m. So you can think of this as a very special formula for functions of the type e to the constant times x. Where does this function fit in? it turns out that functions of they type a to the x, can always be written as e to the m times x. 2, an identity that involves exponential functions, 2 can be written as e to the ln2.
Any number 3 is e to the ln3. So you can always use this identity to rewrite this function, with base e. So you’d get the derivative, this is the same as the derivative of e to the natural log 2 times x. These are the same function.
So when I differentiate this function keeping in mind that ln2 is just a constant, I get e to the ln2 times x times ln2. And because this is the same as 2 to the x, I can write 2 to the x times ln2. And you recall that this is the formula that we had before we differentiated 2 to the x. This is how you can derive the formula. Just change the 2 to e to ln2, and that’s how you differentiate any exponential function.