1. The CHAIN Rule The first of the really cool things that we learn in Calculus

2. Remember how you convert units…  If a BB gun has a muzzle velocity (speed that the BB exits the muzzle) of 500 feet per second, what is the muzzle velocity in the more familiar unit of miles per hour?

3. Same idea…  In the same way that we could convert feet per second to miles per hour by using unit multipliers, we’re going to rewrite some functions as composition of functions and find the derivatives by creating a “chain”

4. Example #1…  Let’s start off with an easy one  f(x) = (2x+1) 2  We can assign 2x+1 to be a and then define f(x) as a 2  And then we could find f’(x) by finding the derivative of a with respect to x and the derivative of f(x) with respect to a; or, symbolically

5. Now you try one…  If h(x) =  Define a function a  Define h(x) in terms of a  Define your derivatives  Create your “chain” and simplify

6. Example #2…  Let’s get a little harder g(x) =  Let’s assign a to be x + 2 and the g(x) as or a -½

7. Now let’s try a tougher one… j(x)=x 2 (x – 2) 4 To find j’(x) we’ll use the chain rule AND the product rule

8. Now you try one…  If m(x) =  Express m’(x) using the quotient rule  Define the derivatives that are immediately definable  Create your necessary “chain” and simplify

9. One more tough one (which is, admittedly, a pretty good jump)… k(x)= To find k’(x) we’ll use a series of chains

10. Same problem with a slightly different path… k(x)= Rather than defining k(x) from the simple to complex, we can go the other way…

11. Now you try one with multiple “chains”…  If p(x) = ((x 2 + 3) 5 + x) 2  Create a chain that allows you to express p’(x)  Define the derivatives with respect to the variables used for each expression  Create your necessary “chain” and simplify