Once a calculus student learns the chain rule, they need to use it virtually all the time. It’s like learning to read when you’re a kid: \u00a0once you know how, you’re expected to do it forever.<\/p>\n
The chain rule is a quintessential technique of calculus: \u00a0it allows you to easily find derivatives you could barely hope to accomplish any other way. \u00a0Its fundamental meaning is as follows:<\/p>\n
Let’s imagine you have two functions, f(x) and g(x). They have been composed into the function f(g(x)).<\/p>\n
The derivative of f(g(x)) is the derivative of the outer one, still at g(x), multiplied by the derivative of the inner one. Behold:<\/p>\n
d\/dx f(g(x))=f'(g(x))g'(x)<\/p>\n
Example 1:<\/strong><\/p>\n Find the derivative of (3x^2-7)^11.<\/p>\n Without the chain rule, this derivative would be hopeless to attempt. With the chain rule, it’s straightforward.<\/p>\n Step 1: Realize which is the outer function versus which is the inner function.<\/p>\n In this case, the outer function is f(x)=(g(x))^11. The inner function is g(x)=3x^2-7<\/p>\n Step 2: Find the derivative of the outer function.<\/p>\n d\/dx (g(x))^11 = 11(g(x))^10<\/p>\n Step 3: Find the derivative of the inner function.<\/p>\n The derivative of the inner function is easily found as well, using the power rule and the constant rules:<\/p>\n d\/dx (3x^2-7)=6x<\/p>\n Step 4: Multiply the outer and inner derivatives to get the final answer:<\/p>\n d\/dx (3x^2-7)^11 = (11(3x^2-7)^10)*6x<\/p>\n You may now want to simplify:<\/p>\n d\/dx =66x(3x^2-7)^10<\/p>\n Example 2:<\/strong><\/p>\n Find the derivative of f(x)=cos(x^7)<\/p>\n Solution:<\/p>\n Step 1: Obviously, the outer function is cos(inner function); the inner function is x^7.<\/p>\n Step 2: We find the derivative of the outer function:<\/p>\n (cos(inner function))’=-sin(inner function)=-sin(x^7)<\/p>\n Step 3: We find the derivative of the inner function:<\/p>\n (x^7)’=7x^6<\/p>\n Step 4: We multiply the outer and inner derivatives together:<\/p>\n -sin(x^7)*7x^6<\/p>\n You may now want to simplify:<\/p>\n -7x^6sin(x^7)<\/p>\n I’ll be talking more about the chain rule in future posts. HTH:)<\/p>\n Source:<\/p>\n Larson & Hostetler: Calculus, part one<\/em>, third ed. Toronto: D.C. Heath and Company, 1989.<\/p>\n