Differentiation of Exponentials and Logs: Exercise 2

00:01 Let's build this up now. We'll look at our second example, we now have a function y equals to e to the 2x plus 1, plus 5, all to the power of 4.

00:13 Now this is getting a little bit more exciting now, let's look at this and break this function down.

00:19 You have a function or something to the power of 4.

00:23 You then have a function within a function inside of it, so we also have this and we also have this.

00:29 Let's ignore the inside function for now.

00:31 Remember what we said, as soon as you spot there is something is a function within a function, we start with the outside, we ignore everything that's on the inside and then we simply differentiate.

00:42 So when we start to dy/dx, I bring the power to the front, so that gives me 4.

00:49 The inside stays exactly as it is for now, plus 5 and then you decrease the power by 1, so I have done, I'm done with differentiating the outside function, we now look at the inside function and we start to differentiate this.

01:04 So e to the power of 2x plus 1, if I do that here so e to the power of 2x plus 1, remember that in itself is a function of a function again, so we have an e function and then we have another function inside of it.

01:22 In order to differentiate it, remember e to the power of anything stays as it is, so that doesn't change.

01:28 And then the differential of 2x plus 1 is just 2.

01:32 So coming back to this little part here, e to the 2x plus 1 is 2e to the 2x, plus 1.

01:41 And the 5 is just the constant so that disappears.

01:44 Let's put it all together, we have 4 times 2 so the numbers can multiply, I have 8e to the 2x plus 1 and then I have the brackets, e to the 2x plus 1, plus 5 all to the power of 3. And that is the derivative of e to the 2x plus 1 plus 5 all to the power of 4.

02:06 So this is fairly interesting because this almost had a function inside of function, and then a function of a function inside of it.

02:12 So you can make them as complicated as you want, but you just break it down step by step.