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Chain Rule: The General Power Rule - Concept

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.

The general power rule is a special case of the chain rule. It is useful when finding the derivative of a function that is raised to the nth power. The general power rule states that this derivative is n times the function raised to the (n-1)th power times the derivative of the function.

The chain rule is one of the toughest topics in Calculus and so don't feel bad if you're having trouble with it. Because it's so tough I've divided up the chain rule to a bunch of sort of sub-topics and I want to deal with a bunch of special cases of the chain rule, and this one is going to be called the general power rule. We'll get into that in a second, the call with the chain rule is it's a method for differentiating composite functions like f of g of x and I've been in a habit of color coding my composite functions so that the inside part is blue and the outside part is red. Anyway this is the chain rule I want to introduce you to what I'm calling a general power function so h of x is the general power function if it could be written as some function g of x any function raised to the nth power. So how do you differentiate one these well we're going to use a version of the chain rule that I'm calling the general power rule. So the derivative of g of x to the n is n times g of x to the n minus 1 times the derivative of g of x.
This is just a special case of the chain rule, so let's try it out on this function h of x equals this function 2x cubed plus 3x-1 all raised to the negative 7 power. And so just to be absolutely clear I'm going to color code this function so that we can see what's the inside function and what's the outside function. Usually the best way to identify inside versus outside is to think about calculating values. Like if I were going to plug 5 into this function I'd raise 5 to the third power multiply by 2 add 3 times 5, I'd be working on this part of the function. So this is the inside function, so equals and the outside function is the raising to the negative 7 that's the outside function and inside I'll make blue 2x cubed plus 3x-1. So according to this rule h prime is going to be the derivative of this and so I take this n the exponent pull out in front so I get negative 7 times this quantity 2x cubed plus 3x-1.
And the new exponent is going to be the old exponent minus 1, so negative 7-1 is negative 8 times and then the derivative of the inside function and that's going to be 6x squared plus 3 I'll put that here and your teacher may want you to simplify this. And so let's make the observation that this quantity here because I have a negative 8 exponent it's going to end up in the denominator. So I have a fraction and I'll have 2x cubed plus 3x-1 to the eighth power. And in the numerator I'll have negative 7 times 6x squared plus 3. That's negative 42 x squared oops minus because I have to distribute the negative 7 over these 2 terms so negative 7 times 3 is negative 21 and that's my answer. So don't forget the general power rule is just a specific special case of the chain rule. Derivative of g of x to the n is n g of x to the n-1 times g prime of x.

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