1. AP Calculus Chapter 3 Section 3
Rules for Differentiation

2. Rule 1: Derivative of a Constant
The first rule of differentiation is that the derivative of every constant function is the zero of the function.
If f(x) = c, then
Proof:

3. Rule 2: Power Rule for Positive Integer Powers of x
If n is a positive integer, then
The proof is on page 116.

4. Rule 2: Examples

5. Rule 3: Constant Multiple Rule
If u is a differentiable function of x and c is a constant, then
Example:

6. Rule 4: Sum & Difference Rule
If u and v are differentiable functions of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such points,
Example:
This is a very powerful rule because it allows us to find the derivatives of polynomials.

7. Rule 5: Product Rule
The product of 2 differentiable functions u and v is differentiable, and
“the first times the derivative of the second plus the second times the derivative of the first”
Note: the order you do the differentiation does not matter because we are adding the two together.

8. Product Rule Example Find the derivative of 𝑓 𝑥 =( 𝑥 2 +1)( 𝑥 3 +3)
Solution: Let 𝑢= 𝑥 & 𝑣= 𝑥 3 +3 , then
𝒅 𝒅𝒙 ( 𝒙 𝟐 +𝟏)( 𝒙 𝟑 +𝟑) = 𝒙 𝟐 +𝟏 𝟑 𝒙 𝟐 + 𝒙 𝟑 +𝟑 𝟐𝒙
=𝟑 𝒙 𝟒 +𝟑 𝒙 𝟐 +𝟐 𝒙 𝟒 +𝟔𝒙
=𝟓 𝒙 𝟒 +𝟑 𝒙 𝟐 +𝟔𝒙

9. Rule 6: Quotient Rule
At a point where v ≠ 0, the quotient y = u/v of 2 differentiable functions is differentiable, and
“the bottom times the derivative of the top minus the top times the derivative of the bottom, all over the bottom squared”

10. Quotient Rule Example 𝑓 ′ 𝑥 = 𝑥 2 +1 2𝑥 −( 𝑥 2 −1)(2𝑥) ( 𝑥 2 +1) 2
Differentiate 𝑓 𝑥 = 𝑥 2 −1 𝑥 2 +1
Solution: Let 𝑢= 𝑥 2 −1 & 𝑣= 𝑥 2 +1, 𝑡ℎ𝑒𝑛
𝑓 ′ 𝑥 = 𝑥 𝑥 −( 𝑥 2 −1)(2𝑥) ( 𝑥 2 +1) 2
= 2 𝑥 3 +2𝑥 −(2 𝑥 3 −2𝑥) ( 𝑥 2 +1) 2
= 4𝑥 ( 𝑥 2 +1) 1

11. Rule 7: Power Rule for Negative Integer Powers of x
If n is a negative integer and x ≠ 0, then
We will use this rule after we discuss the Chain Rule.
This rule can be used to solve for a derivative in place of the Quotient Rule because we can redefine the denominator as a negative integer power of x.

12. Second & Higher Order Derivatives
We can find the derivative of the derivative, which is called the second derivative. Denoted:
We can also find the third, fourth, fifth, and nth derivatives. Denoted:

13. Example
Find the first four derivatives of y = x3 – 5x2 + 2

14. Example
Find the first four derivatives of y = x3 – 5x2 + 2

15. Finding Instantaneous Rate of Change
An orange farmer currently has 200 trees yielding an average of 15 bushels of oranges per tree. She is expanding her farm at the rate of 15 trees per year, while improved husbandry is improving her average annual yield by 1.2 bushels per tree. What is the current (instantaneous) rate of increase of her total annual production of oranges?

16. Solution Let the functions t and y be defined as follows:
𝑡 𝑥 =𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑒𝑒𝑠 𝑥 𝑦𝑒𝑎𝑟𝑠 𝑓𝑟𝑜𝑚 𝑛𝑜𝑤.
𝑦 𝑥 =𝑦𝑖𝑒𝑙𝑑 𝑝𝑒𝑟 𝑡𝑟𝑒𝑒 𝑥 𝑦𝑒𝑎𝑟𝑠 𝑓𝑟𝑜𝑚 𝑛𝑜𝑤.
Then, 𝑝 𝑥 =𝑡 𝑥 𝑦 𝑥 𝑖𝑠 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑜𝑟𝑎𝑛𝑔𝑒𝑠 𝑖𝑛 𝑦𝑒𝑎𝑟 𝑥.
We know: 𝑡 0 =200, 𝑡 ′ 𝑥 =15;𝑦 0 =15, 𝑦 ′ 0 =1.2
We need to find 𝑝 ′ 0 , 𝑤ℎ𝑒𝑟𝑒 𝑝=𝑡𝑦.
Apply Product Rule: 𝑝 ′ 𝑥 =𝑡 0 𝑦 ′ 0 +𝑦(0) 𝑡 ′ (0)
= (15)(15)
=465 𝑏𝑢𝑠ℎ𝑒𝑙𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟

17. Homework
# 3 – 52 by multiples of 3 on pages