 Let's look at the derivatives of logarithmic and exponential functions. So our two key functions are e to the x and log of x, and our derivatives are the following. Derivative of log is 1 over x, again in prime notation, meaning exactly the same thing, and also for e to the x. This is the world's simplest derivative. The derivative of e to the x is e to the x. And again, prime notation e to the x prime is the same as e to the x. And the thing to remember is chain rule, chain rule, chain rule. So let's take the derivative of log of 3x plus 5. And again, we'll start out by a little bit of analysis. So I have my value of x times 3, add 5, hit it with the log, and so this type of function is a log function. So the derivative begins with the derivative of log. And again, I'll drop out everything except for the last function. Derivative of log 1 over times chain rule derivative of whatever was there. And we apply the kindergarten rule, put things back where you found them. And now I need to find the derivative of 3x plus 5, and that's just going to be there. And there's my derivative, which we can simplify a little bit, but we're not going to do too much with. So there's my great simplification. I don't know if I have something that looks more horrendous. Derivative of log of x squared plus 2x minus 7. And what do we have here? Again, take x squared, take x times 2, squared plus 2x, subtract 7, figure that out, then hit the whole thing with the log. This is once again a log function. So I'll drop everything except for that last thing that I do. Derivative of log 1 over times derivative of what was inside. And put things back where you found them. Derivative of x squared plus 2x minus 7, that's just a polynomial. So that's not really a difficult derivative to find. And we don't need to, but a nice simplification is to write that as a single fraction. Again, a useful check to make sure you're applying the derivative rules correctly. Any time you're doing the derivative of anything more complicated than a polynomial, there's always going to be an echo of the original function. So again, here's our x squared plus 2x minus 7 in the original function. Here's our echo in the final function. How about an exponential function, e to the 5x. And again, a little bit of analysis goes a long ways. Take x times 5, e raised to that power. The last thing we do in this function is e raised to the power. So this is an exponential function. Drop everything, except for the last thing that we do. The derivative of e to the is world's easiest derivative. Same thing. Times, don't forget the chain rule. Derivative of whatever was in there. Apply the kindergarten rule. Put things back where you found them. And we do have to find the derivative of 5x, which is just going to be 5. So our echo of the original function, well in this case it's a pretty complete echo, e to the 5x here, e to the 5x down here. And it's really not too much different if we have a more complicated function. Here's e to minus x squared over 2. So again, take x, square it, change the sign, divide by 2, e raised to that number. And that means again we have our last function is e raised to the. And so when we want to find the derivative, we're going to start by dropping everything. This is an e to the type function. The derivative world's easiest derivative. Same thing, times the derivative of whatever was in the parentheses. So put things back where you found them. I need to find the derivative of minus x squared over 2, which is the derivative of minus a half x squared, which is the derivative of, which is minus 1 half times 2x. And I might do a little bit of simplification to make things look a little bit nicer.