Derivative of log and exponential functions

#Derivative of log and exponential functions plus

OK, the slow way would be, well I have a composition of functions here. So, I'll even write down this is the slow way. And then I'll show you why it's good to kind of pull back from a problem sometimes, see how you can make it a lot simpler for yourself, and then solve the problem. There's a fast way and there's a slow way to do this. So that's kind of a nice thing there.ДЄnd then the last one, example three, I'll work out to the right. So it's interesting, now we see that there are trigonometric functions that I can take a derivative of something that's not just trigonometric and get something that's trigonometric. So the reason I, in particular, like this one is that we see, 'Oh, if I wanted to find a function whose derivative was tangent x, a candidate would be the negative of the natural log of cosine of x.' That in fact gives us a function whose derivative is tangent x. So you get this whole thing is negative sine over cosine. And then we take the derivative of the inside function, which is the derivative of cosine x. So we take the derivative of the natural log function, evaluate it at cosine x. So again, what we do is we take the derivative of natural log. But you have seen many times now, when you have natural log of a function, its derivative is going to be 1 over the inside function times then the derivative of the inside function. Because we have a function of a function. So, this is going to require us to do the chain rule. I think it's an interesting function once you find out what the derivative is. So that's natural log of pi times pi to the x. So the derivative of this, we need the rule that we have for derivatives of exponential functions. Because this is not a power of x, this is x is the power. So if you wrote the derivative of this term was x times pi to the x minus one, you would not be alone in the world.Д«ut that is not the correct answer, all the same. It's actually an exponential function right, with base pi.

OK, but the whole point of this problem for me, is to make sure that you recognize that pi to the x is not a power of x rule that needs to be applied. So we can write this as that derivative is, pi times x to the pi minus one.

So the derivative of x to the pi is nice and simple because that is our rule we know for powers of x. OK, now, the reason in particular that I did this one- it might have seemed simple to you, but the reason I did this one is because of a common mistake that people make. OK, so let's start off with the derivative of the first one. And then we'll come back and I will work them out for you as well. I'm going to give you a moment to to work on those and figure those out using the the tools you now have. So you have three functions you want to take the derivative of with respect to x. And the third one is- that's an h not a natural log- h of x is equal to natural log of e to the x squared. The second function is g of x is equal to natural log of cosine of x.

#Derivative of log and exponential functions plus

The first one is f of x is equal to x to the pi plus pi to the x. And I'd like us to find derivatives of the following functions. So I have three particular examples that I want us to look at. We're going to practice using some of the tools you developed recently on taking derivatives of exponential functions and taking derivatives of logarithmic functions. CHRISTINE BREINER: Welcome back to recitation.