Chain Rule: The General Power Rule - Problem 2
We’re using a special case of the chain rule that I call the general power rule. Here’s a problem that we can use it on. Differentiate y equals x² times the square root of x² minus 9. Whenever I’m differentiating a function that involves the square root I usually rewrite it as rising to the ½ power.
This is going to be x² times x² minus 9 to the ½ power. Because I’m going to use the general power rule on this piece of the function. But notice this part of the function is also a product. So I’m going to have to use the product rule. Just making sure that you see this; this left piece is one of the factors in the product and this right piece is another. I’ll use the general power rule on this piece but I don’t need the general power I just need the regular power rule for this guy.
Let’s differentiate. We have dy/dx equals, now I’m using the product rule here; the first times the derivative of the second. The first was x² and then we want the derivative of, and I’m going to, I’m not going to actually do the derivative I just want to represent it by using this notation. The derivative of x² minus 9 to the ½ plus the second x² minus 9 to the ½, times the derivative of the first.
This step is just the product rule. I haven’t actually done any actual derivatives yet. Next, x² times, and here’s where I need the general power rule. The ½ comes in front and I have (x² minus 9) to the -1/2. Replace this exponent with 1 less. So -1/2 times the derivative of the inside function and that’s the derivative of this. It’s going to give me 2x. Plus and now I’m on this term, x² minus 9 to the ½ times the derivative of x², 2x.
Let’s see how this combines. We have a ½ and a 2 which shall cancel. So these cancel. I have an x and an x ², that makes x ³, and this x² minus 9 to the ½, it’s like 1 over x² minus 9 to the +½. So that becomes the square root of x² minus 9 in the denominator. And over here, I’ve got 2x times the square root of x² minus 9. And that’s my derivative of the function y equals x² times root x² minus 9.