Chain Rule

# Chain Rule - Chain Rule Weve taken a lot of derivatives...

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Chain Rule We’ve taken a lot of derivatives over the course of the last few sections. However, if you look back they have all been functions similar to the following kinds of functions. These are all fairly simple functions in that wherever the variable appears it is by itself. What about functions like the following, None of our rules will work on these functions and yet some of these functions are closer to the derivatives that we’re liable to run into than the functions in the first set. Let’s take the first one for example. Back in the section on the definition of the derivative we actually used the definition to compute this derivative. In that section we found that, If we were to just use the power rule on this we would get,

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which is not the derivative that we computed using the definition. It is close, but it’s not the same. So, the power rule alone simply won’t work to get the derivative here. Let’s keep looking at this function and note that if we define, then we can write the function as a composition. and it turns out that it’s actually fairly simple to differentiate a function composition using the Chain Rule . There are two forms of the chain rule. Here they are. Chain Rule Suppose that we have two functions f(x) and g(x) and they are both differentiable. 1. If we define then the derivative of F(x) is, 2. If we have and then the derivative of y is, Each of these forms have their uses, however we will work mostly with the first form in this class. To see the proof of the Chain Rule see the Proof of Various Derivative Formulas section of the Extras chapter. Now, let’s go back and use the Chain Rule on the function that we used when we opened this section.
Example 1 Use the Chain Rule to differentiate . Solution We’ve already identified the two functions that we needed for the composition, but let’s write them back down anyway and take their derivatives. So, using the chain rule we get,

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And this is what we got using the definition of the derivative. In general we don’t really do all the composition stuff in using the Chain Rule. That can get a little complicated and in fact obscures the fact that there is a quick and easy way of remembering the chain rule that doesn’t require us to think in terms of function composition. Let’s take the function from the previous example and rewrite it slightly. This function has an “inside function” and an “outside function”. The outside function is the square root or the exponent of depending on how you want to think of it and the inside function is the stuff that we’re taking the square root of or raising to the , again depending on how you want to look at it. The derivative is then,
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## This note was uploaded on 11/06/2011 for the course MATH 151 taught by Professor Sc during the Fall '08 term at Rutgers.

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Chain Rule - Chain Rule Weve taken a lot of derivatives...

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