# CHAIN RULE Essay

504 WordsAug 18, 20083 Pages
First, let&#8217;s review composite functions. A composite function is the combination of two functions, usually written as (f o g)(x) or f(g(x)). The best way to understand composite functions is to envision one as a series of two machines. Do not confuse a composite function with the product of two functions. An example of a composite function is h(x) = sin(5x), where f(x)=sin(g(x)) and g(x) = 5x. Now, let&#8217;s ask ourselves: How would we differentiate this function? Other composite functions like h(x) = (x+1)5 can be differentiated by expanding first and then applying the power rule, but that is a long-winded process. Luckily, the process becomes shorter when you use binomial expansion, but what about the function, h(x) = (x3 + x2 + x + 1)5? You can&#8217;t use binomial expansion, so it becomes extremely tedious. And in the case of h(x) = sin(5x), as well as in the case of h(x) = (ex + ln(x))7, you cannot simply rely simply on rules you know already, like the power rule and the rules for differentiating trigonometric, exponential, and logarithmic functions. You need to use the &#8220;chain rule.&#8221; The Chain Rule: If f is differentiable at the point u = g(x), and g is differentiable at x, then the composite function (f o g)(x) = f(g(x)) is differentiable at x, and (f o g)&#8242;(x) = f&#8242; (g(x)) &#61620; g&#8242;(x) In Leibniz notation, if y = f(u) and u = g(x), then where dy/du is evaluated at u = g(x). Now that we know the chain rule, let&#8217;s apply it to the function h(x) = sin(5x) and find its derivative at x = 2&#960;/5. You can use the chain rule repeatedly like in the case of g(t) = tan(5 &#8211; sin(2t)). Find its derivative. ------------------------------------------------------------------------------------------------------------ The chain rule now presents us the opportunity to integrate