Recall that a composite function f(g(x)) is a function that has another function on the "inside." When taking the derivative of a function like this, we use the chain rule. The chain rule states that you first take the derivative of the "outside" function, then multiply it by the derivative of the "inside function." So for a function h(x)=f(g(x)), its derivative would be h'(x)=f'(g(x))*g'(x).
To determine which function is the inside function, look to see which function is "contained" within another function. For example, for exponential functions, look at the power to which e is raised. For logarithmic functions, it will be what is within the logarithm brackets.
Let’s take a look at the chain rule again. I want to differentiate a function h(x) equals the natural log of x² plus 4x plus 5. So here I have to identify the inside function and the outside function.
It might be clear to you when a function’s written this way, that the inside function is the function that is literally inside the parenthesis. And that’s right. So x² plus 4x plus 5 is going to be the inside function. I’ll color code that in blue. X² plus 4x plus 5. The outside function I’ll color in red. So natural log of all that stuff.
When we’re differentiating, h'; we’re going to differentiate the outside function first. So f', now the derivative of natural log, natural log x is 1 over x. So we want the derivative of natural log evaluated at all these stuff. So 1 over all of this. That’s the derivative of natural log. 1 over, I will write it in blue still, x² plus 4x plus 5.
I have to multiply that by the derivative of the inside function. That’s this part. The derivative is going to be (2x plus 4). So this is exactly the same as (2x plus 4) over x² plus 4x plus 5. That’s the derivative of natural log of x² plus 4x plus 5.