Chain Rule: The General Logarithm Rule - Problem 1


You can use the chain rule to find the derivative of a composite function involving natural logs, as well. Recall that the derivative of ln(x) is 1/x. For example, say f(x)=ln(g(x)), where g(x) is some other function of x. By the chain rule, take the derivative of the "outside" function and multiply it by the derivative of the "inside" function. With the derivative of logarithmic functions, the outside function is the logarithm itself, and the inside function is what is inside the logarithm. So, f'(x)=1/g(x) * g'(x).


We’re talking about how to differentiate a special kind of composite function. This kind; natural log of g(x). Now because the derivative of natural log is 1 over x, the derivative quantity of the chain, would be 1 over g(x) times g'(x). So let’s differentiate one of these kinds of functions. H(x) equals natural log of 1 plus e to the x.

I have it written up here; h' is going to be 1 over the inside stuff, 1 over 1 plus e to the x times the derivative, with respect to x of 1 plus e to the x. The derivative of the inside. Of course that’s just going to be zero plus e to the x. So 1 plus e to the x here. And that’s just e to the x over 1 plus e to the x. that’s the derivative of natural log of 1 plus e to the x.

chain rule logarithmic functions derivative of natural log exponential functions