Chain Rule: The General Power Rule - Problem 3
Let’s do a harder example. I want to differentiate h(x) equals 64x to the 6th plus 2 over the quantity x to the 4th plus 1 to the 6th power. Let me use the general power rule but I’m only going to use it when I’m differentiating the denominator. But overall this function’s a quotient. So I’m going to need to need the quotient.
Let’s get started with the derivative. So h'(x) equals low d high, so I have x to the 4th plus 1 to the 6th, times the derivative of the numerator. That’s going to be 6 times 64, 384x to the 5th. Minus high d low, that’s going to be 64, x to the 6th plus 2 times the derivative of the denominator. For the derivative of the denominator I do need to use the general power rule. So the 6 is going to come in front, I have 6 (x to the 4th plus 1) and the exponent drops by 1 so 5. I have to multiply by the derivative of x to the 4th plus 1. That’s 4x³.
In the denominator, I get the square of what’s below, so the x to the 4th plus 1 to the 6th power becomes to the 12th power. And this is the raw form of the derivative; I just need to do a lot of simplification.
First let’s observe that this x to the 4th plus 1 appears in both of the terms of the numerator and the denominator. So some cancellation is going to take place. Let me do that with a different color. I have 5 of them here, 6 of them here and 12 down here. So I can cancel 5 between numerator and denominator. This one’s completely gone, 5 of these are gone, leaving one. And then 5 these are gone leaving 7.
I distribute this 384x to the 5th over the one remaining factor. I get 384x to the 9th plus 384x to the 5th. That’s take care of this part. Over here this factor’s now gone. I have 24x³ on the outside and I have to distribute this over these two terms. That’s all I have left 24x³.
24 times 64 is 1536. This is going to be minus, -1536x to the 9th. And then I have 2 times 24x³ and that’s going to be negative. So -48x³ all over x to the 4th plus 1 to the 7th power. All I have to do is combine like terms and I’ll be done. So 384x to the 9th minus 1536x to the 9th is -1152x to the 9th. I have an x to the 5th term, 384x to the 5th and -48x³. All that over (x to the 4th + 1) to the 7th power.
There’s a lot of simplification but usually your teacher will want you to simplify answers like this. Take a look at this function, it actually looks like Batman if you graph it out. Here is its complicated derivative; -1152x to the 9th plus 384x to the 5th minus 48x³ all over x to the 4th plus 1 to the 7th power.