![]() The proof of the chain rule is found in Appendix E.4. 3.6.4 Recognize the chain rule for a composition of three or more functions. ![]() 3.6.3 Apply the chain rule and the product/quotient rules correctly in combination when both are necessary. 3.6.2 Apply the chain rule together with the power rule. Example: Suppose you are driving westward across Colorado on IH 70 (to get to Vail to go snowskiing). 3.6.1 State the chain rule for the composition of two functions. Graphically, this means that the max/min value is the maximum/minimum height of the graph at some x c. The proof of the chain rule uses this essential idea, but care is taken to ensure that the quantity \(\Delta u\) is nonzero, to avoid the embarrassment of dealing with the nonsensical ratio 0/0. extrema Watch on Notice that the absolute and local maxima and minima are y -values. This agrees with the de nition offtimes the derivative ofg. Similarly, we can use the chain rule to find the partial derivative fy. In the case of finding global extremes over the functions entire domain, we again use a first or second derivative sign chart. Step 3: Determine the derivative of the outer function, dropping the inner function. optimization and finding local extreme points. Step 2: Know the inner function and the outer function respectively. Example: For linear functionsf(x) ax b g(x) cx d, the chain rule canreadily be checked: we havef(g(x)) a(cx d) bacx ad bwhich has thederivativeac. Steps to Obtain Chain Rule Step 1: Recognize the chain rule: The function needs to be a composite function, which implies one function is nested over the other one. ![]() Then it is apparent that the "cancellation" of terms \(\Delta u\) in numerator and denominator lead to the correct fraction on the left. Solution: applyingthe chain rule gives cos( cos(x)) ( sin(x)).
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