1. 2.5
The Chain Rule

2. The Chain Rule

3. Example Find F '(x) if F (x) = .
Solution: (using the first definition)
F (x) = (f  g)(x) = f (g(x)) where f (u) =
and g (x) = x2 + 1.
Since and g(x) = 2x
we have F (x) = f (g (x))  g (x)

4. Solution: (using the second definition)
let u = x2 + 1 and y = , then

5. The Chain Rule for powers:

6. Example Differentiate: y = (x3 – 1)100. Solution:
Taking u = g(x) = x3 – 1 and n = 100
= (x3 – 1)100
= 100(x3 – 1) (x3 – 1)
= 100(x3 – 1)99  3x2
= 300x2(x3 – 1)99

7. Practice:

8. Implicit Differentiation
2.6
Implicit Differentiation

9. Implicit Differentiation
So far we worked with functions where one variable is expressed in terms of another variable—for example:
y = or y = x sin x (in general: y = f (x). )
Some functions, however, are defined implicitly by a relation between x and y, examples:
x2 + y2 = 25
x3 + y3 = 6xy
We say that f is a function defined implicitly -
For example Equation 2 above means: x3 + [f (x)]3 = 6x f (x)

10. Example 1
For x3 + y3 = 6xy find:

11. Example 2
Find the equation of the tangent line at (3,2)

12. Answer:

13. Practice problem:
Find the slope of the curve at (4,4)

14. Practice:
In each case, find y’