2.5 The Chain Rule

The Chain Rule

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)

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

The Chain Rule for powers:

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

Practice:

Implicit Differentiation 2.6 Implicit Differentiation

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)

Example 1 For x3 + y3 = 6xy find:

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

Answer:

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

Practice: In each case, find y’