Derivatives, Properties, and Formulas
Read this section to understand the properties of derivatives. Work through practice problems 111.
Example 4: The derivative of is , and the derivative of is
What are the derivatives of their elementary combinations: and ?
Solution:
Unfortunately, the derivatives of and don't follow the same easy patterns:
These two very simple functions show that, in general, and .
The Main Differentiation Theorem below states the correct patterns for differentiating products and quotients.
Practice 2: For and , what are the derivatives of and ?
The following theorem says that the simple patterns in the example for constant multiples of functions and sums and differences of functions are true for all differentiable functions. It also includes the correct patterns for derivatives of products and quotients of differentiable functions.
Main Differentiation Theorem: If and are differentiable at , then
(a) Constant Multiple Rule: or
The proofs of parts (a), (b), and (c) of this theorem are straightforward, but parts (d) and (e) require some clever algebraic manipulations. Lets look at an example first.
Example 5: Recall that and . Find and .
Solution: is an application of part (a) of the theorem with and so
uses part (c) of the theorem with and so
Practice 4: Fill in the values in the table for , and
0 
3 
–2 
–4 
3 



1 
2 
–1 
1 
0 



2 
4 
2 
3 
1 



Proof of the Main Derivative Theorem (a) and (c): The only general fact we have about derivatives is the definition as a limit, so our proofs here will have to recast derivatives as limits and then use some results about limits. The proofs are applications of the definition of the derivative and results about limits.
The proof of part (b) is very similar to these two proofs, and is left for you as the next Practice Problem.
The proof for the Product Rule and Quotient Rules will be given later.
Practice 5: Prove part (b) of the theorem, the Sum Rule: .
Practice 6: Use the Main Differentiation Theorem and the values in the table to fill in the rest of the table.
0 
3 
–2 
–4 
3 



1 
2 
–1 
1 
0 



2 
4 
2 
3 
1 



Solution: (a) We can use the Product Rule with and :
(b) We can use the Quotient Rule with and :
Proof of the Product Rule: The proofs of parts (d) and (e) of the theorem are complicated but only involve elementary techniques, used in just the right way. Sometimes we will omit such computational proofs, but the Product and Quotient Rules are fundamental techniques you will need hundreds of times.
By the hypothesis, and are differentiable so and .
Also, both and are continuous (why?) so and .
(d) Product Rule: Let . Then .
finding common factors  
taking limit as  
(e) The steps for a proof of the Quotient Rule are shown in Problem 55.