Differentiating

1
Differentiating Combined Functions

• Deriving the Sum and Product Rules for
  Differentiation.

2
Algebraic Combinations

• We have seen that it is fairly easy to compute
  the derivative of a simple function using the
  definition of the derivative.
• More complicated functions can be difficult or
  impossible to differentiate using this method.
• So we ask . . . If we know the derivatives of
  two fairly simple functions, can we deduce the
  derivative of some algebraic combination (e.g.
  the sum or product) of these functions without
  going back to the difference quotient?

Yes!
3
The Limit of the Sum of Two Functions
4
The Limit of the Product of Two Functions
5
Continuity Required!
In the course of the calculation above, we said
that Is this actually true? Is it ALWAYS
true?
6
The Limit of the Product of Two Functions
Would be zero if g were continuous at xa. Is it?
7
The Limit of the Product of Two Functions
g is continuous at xa because g is
differentiable at xa.