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The Chain Rule - Problem 2

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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 solve another problem with the chain rule. Here I’m asked to differentiate h(x) equals the square root of 4x² plus 9. When I’m using the chain rule, I want to identify what function is the inside function and what function’s the outside function.

I’ve got my h(x). And again when you are trying to identify the inside function think of what you’d have to do first if you were evaluating this function. So if I was plugging in say x equals 1, I’d first square the 1, and get 1, times 4, plus 9. So clearly this polynomial’s the inside function. It’s the first thing you evaluate when you’re evaluating h(x).

I’m going to make that one blue. Blue is my color code for the inside function. So 4x² plus 9. And then the last thing I do is, take the square root. That’s going to be in red. What this means is that I’m going to have to differentiate the square root function and the inside function.

Usually when I differentiate the square root function, I think of it as x to the ½. So it might be a good idea to write this as something to the ½ instead of the square root. Just for the purposes of differentiation, I like to do this. So 4x² plus 9 all raised to the ½ power. That’s h(x). So h'(x), the derivative, is going to be the derivative of the outside function, with the inside left alone, and that’s going to be ½, the ½ comes out in front. Something to the ½ minus 1, -1 ½ . So the inside function is left alone. 4x² plus 9.

You do differentiate the inside function but you don’t do it here. You multiply by that on the outside. That’s what this part of the chain rule says. Multiply by the derivative of 4x² plus 9 and you get 8x right here.

And we can simplify this a little bit. This is going to give us, let me take up here, h'(x), 1/2 and 8x is going to give me 4x. And this -1/2 means I have the square root in the denominator. So that’s going to be one over the square root of 4x² plus 9. What’s on the top? 4x. What’s on the bottom? Root 4x² plus 9. That’s the derivative of h(x).

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