1. 3.4 - The Chain Rule

2. The Chain Rule: Defined If f and g are both differentiable and F = f ◦ g is the composite function defined by F(x) = f(g(x)), then F is differentiable and F´ is given by the product F´(x)=f ´(g(x)) · g´(x) In Leibniz notation, if y = f(u) and u = g(x) are both differentiable functions, then

3. Guided Example 1 The hard part of the chain rule is identifying the compositions being made. Take y = (x 2 – 3x) 10 for example. It is the composition, f(g(x)), of two functions. What are the functions? f(x) = ____________ g(x) = ____________ Substitution Variable FunctionFunction With Substitution Variable Derivative With Respect To Sub Variable y(x 2 – 3x) 10 u 10 d__/d__ = ux 2 – 3x d__/d__ =

4. Guided Example 1 Now apply the chain rule.

5. The Chain Rule: Defined What would the chain rule look like if you had F = f ◦ g ◦ h = f(g(h(x))), the composition of three differentiable functions? F´(x)= ______________________ In Leibniz notation, if y = f(u), u = g(v), and v = h(x) are both differentiable functions, then

6. Guided Example 2 This can get a little tricky when dealing with the composite of several functions. Take y = sin (cos (x 4 )) for example. It is the composition of three functions f (g(h(x))). What are f, g, and h? f(x) = _________, g(x) = ___________, h(x) = ___________ Substitution Variable FunctionFunction With Substitution Variable Derivative With Respect To Sub Variable ysin(cos(x 4 ))d__/d__ = u vx4x4

7. Guided Example 2 Now apply the chain rule.

8. The Chain Rule with the Power Rule If n is any real number and u = g(x) is differentiable, then Alternatively,

9. As you can see, when doing the chain rule in your head, you start from the outside and work your way in. Keep in mind that this is no different than using the substitution method. Doing The Chain Rule In Your Head

10. Guided Example 1 – My Method Easier to do in your head.

11. Guided Example 2 – My Method

12. You Try It Additional Practice with Guided Solutions can be found at http://archives.math.utk.edu/visual.calculus/2/chain_rule.4/index.html Find the first derivative of each.