What is the difference between chain rule and power rule
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 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 And the new exponent is going to be the old exponent minus 1, so negative 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. All Calculus videos Unit Techniques of Differentiation. Learn more.
How do you know when to use the Chain Rule instead of the Power Rule? Asked 5 years, 1 month ago. Active 5 years, 1 month ago. Viewed 6k times. Pineapple29 Pineapple29 2 2 silver badges 10 10 bronze badges. So even if you've "made the power rule problem into a chain rule problem" , you're still using the power rule within your implementation of the chain rule.
Show 4 more comments. Active Oldest Votes. Ben Grossmann Ben Grossmann k 12 12 gold badges silver badges bronze badges. My uncertainty is over whether my statement answers your question, since it isn't clear what exactly you're asking for. Add a comment. MathematicsStudent MathematicsStudent But that's why we have power rule, as a shortcut.
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