Calculus 220 section 3 2 The Chain Rule notes prepared by Tim Pilachowski Composition of functions is taking one function s formula and inserting it into another s Vocabulary and notation varies f composition g f of g f o g f o g x f g x Example A Given f x x 2 x and g x 3 x 1 find the algebraic rule for f o g x Answer f o g 9 x 2 3x To find the derivative of a composition of functions we have the chain rule d f g x f g x g x dx Example A extended Given f x x 2 x and g x 3 x 1 find f g x Answer 18x 3 2 6 2 x 2 x 4 x 1 Example B Given f x 4 x 3 and g x 2 x x find f g x answer d f g x 3 2 dx 3 4 2x 2 x 3 For the time being the only functions we ll be working with are power functions but we ll soon move on to exponential and logarithm functions for which the chain rule is not only important but necessary One way to think of the chain rule is the derivative of the outside applied to the inside times the derivative of the inside Possibly the most important task in using the chain rule is correctly identifying the outside and inside functions i e which one is f and which one is g Section 3 2 is practice in doing precisely this 3 Example C Determine whether h x x 2x has any extrema either relative or absolute Answer absolute minimum at 0 0 Note that h x 0 for all values of x in the domain which means that h x is increasing for all values of x in its domain Since the domain of h 0 we conclude that h has an absolute minimum at x 0 and that it has no maximum We have an alternate way of writing the chain rule using Leibnitz notation for y f u and u g x dy dy du dx du dx The text has a verification of the chain rule based on the original formula given above Here is another way to look at the chain rule using the definition of derivative and properties of fractions u dy y y y u y u lim lim lim lim lim dx x 0 x x 0 x u x 0 u x x 0 u x 0 x Since u is a function of x so as x 0 u also approaches 0 So dy y u y u dy du lim lim lim lim dx x 0 u x 0 x u 0 u x 0 x du dx There are a couple of caveats 1 Both the text s process and the one above assumed u 0 The case in which u 0 requires a good bit of high powered finagling which you will not need to know about 2 As noted earlier in the course the differentials above are not fractions although the notation used can make it appear that way In this case however the form provides a way to remember the chain rule 9 8 2x 1 dy dy Example D Given y 3x 2 2 find Answer 9 3x 2 2 x 2 1 6 x 22 23 dx x x x dx x Example E An environmental study determined that the level of carbon monoxide in parts per million in the air surrounding a small city was a function of the number of people living there p population in thousands C p 0 5 p 2 17 Population is in turn a function of time in years p t 3 1 0 1t 2 dC dp dC and represent b Find the equation to express the level of CO as a function of a Explain what dp dt dt 2 dC Answer b C t 0 5 3 1 0 1t 2 17 c 0 24 time c Use the chain rule to find dt t 3