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.
Now this is getting a little bit more exciting now, let's look at this and break this function down.
You have a function or something to the power of 4.
You then have a function within a function inside of it, so we also have this and we also have this.
Let's ignore the inside function for now.
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.
So when we start to dy/dx, I bring the power to the front, so that gives me 4.
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.
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.
In order to differentiate it, remember e to the power of anything stays as it is, so that doesn't change.
And then the differential of 2x plus 1 is just 2.
So coming back to this little part here, e to the 2x plus 1 is 2e to the 2x, plus 1.
And the 5 is just the constant so that disappears.
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.
So this is fairly interesting because this almost had a function inside of function,
and then a function of a function inside of it.
So you can make them as complicated as you want,
but you just break it down step by step.