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The Chain Rule - Problem 1

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.

For example, let h(x)=e-5x2-6. The "outside" function is ex and the "inside" function is -5x2-6. First, take the derivative of the outside function. Remember that the derivative of ex is itself, ex. So, the first part of our derivative is e-5x2-6. Then, find the derivative of the inside function, -5x2-6. From the power rule, we know that its derivative is -10x. Multiplying these together, the result is h'(x)=-10xe-5x2-6.

Let’s do a problem that involves the chain rule. Let’s recall that the chain rule is a method for differentiating composite functions. And to highlight what a composite function is and how to put together, I’ve color coded the inside part and the outside part.

Remember a composite function is like when you put a function inside another. So here the inside function is g(x) and the outside function is f(x). When you differentiate these things, you differentiate the outside function first. Leave the inside function alone, and then multiply by the derivative of the inside function.

One of the things you have to do when you’re solving a problem is you have to identify the outside and inside functions. Here I’m asked to differentiate h(x) equals e to the 2x³ minus 5. Let’s identify the inside and outside function.

We have our exponential function; e to the x, and then the 2x³ minus 5. One of these is the inside; one of this is the outside function. I think the way I would identify the inside function, is think about if you were computing values for this function. What would you do first? If you had an x value like say 2, you first cube it, multiply by 2 and subtract 5. So it’s clear that this is the inside part of the function.

You could just make the inside function x³. But I think you want to make it all of 2x³ minus 5 because then the outside function would be just e to the x which is nice and easy to differentiate. I’m going to color code this. The outside function is e to the x, I’ll put parenthesis here, or e to the something. And then the inside function is 2x³ minus 5.

Now it’s color coded and it should be really clear how to differentiate h(x). So h'( x) is going to be the derivative of the outside function. The derivative of e to the x is just e to the x. So I’ll write e to the something and the inside function is left alone. 2x³ minus 5 times the derivative of the inside function. That’s what this part means. And the derivative of the inside function is the derivative of this. That’s going to be 6x².

That’s it, that’s the chain rule. Differentiate the outside function first, leave the inside alone and then multiply by the derivative of the inside function.

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