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?

3. The Limit of the Sum of Two Functions

4. The Limit of the Product of Two Functions

5. In the course of the calculation above, we said that Is this actually true? Is it ALWAYS true? Continuity Required! xx+h

6. The Limit of the Product of Two Functions Would be zero if g were continuous at x=a. Is it?

7. The Limit of the Product of Two Functions g is continuous at x=a because g is differentiable at x=a.