1. Basic Differentiation Rules

2. The Constant Rule
𝑑 𝑑𝑥 𝑐 =0
The derivative of a constant function is 0. That is,
Ex 1. 𝑓 𝑥 =7
𝑓 ′ 𝑥 =0
Ex 2. 𝑦=𝜋
𝑦 ′ =0

3. The Power Rule
Let 𝑢=𝑔(𝑥) and 𝑛 be a rational number. Then 𝑑 𝑑𝑥 𝑢 𝑛 =𝑛 𝑢 𝑛−1 ∙𝑢′

4. Ex 3. Find the derivative using the power rule.
c) 𝑦= 1 𝑥 2
b) g 𝑥 = 3 𝑥 −5

5. Ex 4. Find an equation of the tangent line to the graph of 𝑓 𝑥 =2 𝑥 2 +7𝑥−1 at 𝑥=1.

6. Ex 5. Find the derivative of each function.
a) 𝑓 𝑥 = 5 2 x 3
c) 𝑦= 7 3 𝑥 −2
b) 𝑦= 5 2𝑥 3
d) 𝑓 𝑥 = 7 3𝑥 −2

7. e) 𝑓 𝑥 = 3 𝑥 2 −𝑥+1 𝑥
f) 𝑦= 5 𝑥 5 −3 𝑥 4 +4 𝑥

8. Physical Application of the Derivative:
𝑠(𝑡) – position of an object at time 𝑡
𝑣 𝑡 =𝑠′(𝑡) – velocity of an object at time 𝑡
Remark: Differentiate position to get velocity (first derivative).

9. Ex 8. At time 𝑡=0, a diver jumps from a platform diving board that is 32 feet above the water. Because the initial velocity of the diver is 16 feet per second, the position of the diver is 𝑠 𝑡 =−16 𝑡 2 +16𝑡+32, where 𝑠 is measured in feet and 𝑡 is measured in seconds.
When does the diver hit the water?
What is the diver’s velocity at impact?

10. Ex 9. Determine the point(s) at which the given function has a horizontal tangent line.
𝑓 𝑥 = 𝑥 3 −12𝑥