Presentation on theme: "3.6 The Chain Rule Photo by Vickie Kelly, 2002 Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn."— Presentation transcript:
3.6 The Chain Rule Photo by Vickie Kelly, 2002 Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002 U.S.S. Alabama Mobile, Alabama
We now have a pretty good list of shortcuts to find derivatives of simple functions. Of course, many of the functions that we will encounter are not so simple. What is needed is a way to combine derivative rules to evaluate more complicated functions.
Consider a simple composite function:
and one more: This pattern is called the chain rule. dy/dx = 2(3x + 1) 1 3
Chain Rule: If is the composite of and, then: If f(g(x)) is the composite of y = f(u) and u = g(x), then: d / dx (f (g(x) ) = d / dx f ( at g(x) ) d / dx g( at x )
Chain Rule: If is the composite of and, then: example: Find:
We could also find the derivative at x = 2 this way:
Here is a way to find the derivative by seeing layers: Differentiate the outside function, (keep the inner function unchanged...) …then multiply by the derivative of the inner function
Another example: derivative of the outside power function derivative of the inside trig function It looks like we need to use the chain rule again!
Another example: The chain rule can be used more than once. (Thats what makes the chain in the chain rule!)
Each derivative formula will now include the chain rule! et cetera…
The most common mistake in differentiating is to forget to use the chain rule. Every derivative problem could be thought of as a chain-rule situation: derivative of outside function derivative of inside function The derivative of x is one.