9.4 Basic Rules of Differentiation
• Rule 1: Derivative of a Constant
•
𝑑
𝑑𝑥
𝑐 =0
• Rule 3: Derivative of a Constant
Multiple of a Function
•
𝑑
𝑑𝑥
𝑐𝑓 𝑥
=𝑐
𝑑
𝑑𝑥
𝑓 𝑥
• Rule 2:The Power Rule
• If 𝑛 is any real
𝑑
number,
𝑑𝑥
𝑥𝑛
=
𝑛𝑥 𝑛−1
• Rule 4:The Sum Rule
•
𝑑
𝑑𝑥
𝑓 𝑥 ±𝑔 𝑥
=
𝑑
𝑑𝑥
𝑓 𝑥
±
𝑑
𝑑𝑥
𝑔 𝑥
Examples 1-5
• Ex. 1: Find the derivative of the functions.
• (a) 𝑓 𝑥 = 28
(b) 𝑓 𝑥 = 𝜋 2
• Ex. 2: Find the derivative of the functions.
• (a) 𝑓 𝑥 = 𝑥
(b) 𝑓 𝑥 = 𝑥 8
• Ex. 3: Find the derivative of the functions.
• (a) 𝑓 𝑥 = 𝑥
(b) 𝑔 𝑥 =
1
3
𝑥
• Ex. 4: Find the derivative of the functions.
• (a) 𝑓 𝑥 = 5𝑥 3
(b) 𝑓 𝑥 =
3
𝑥
• Ex. 5: Find the derivative of the functions.
• (a) 𝑓 𝑥 = 4𝑥 5 + 3𝑥 4 − 8𝑥 2 + 𝑥 + 3
• (b) 𝑔 𝑡 =
𝑡2
5
+
5
𝑡3
Examples 6-7
• Ex. 6: Find the slope and an equation of the tangent line to the graph of 𝑓 𝑥 = 2𝑥 +
1
𝑥
at the point 1, 3 .
• Ex. 7: A group of marine biologists at the Neptune Institute of Oceanography
recommended that a series of conservation measures be carried out over the next
decade to save a certain species of whale from extinction. After the conservation
measures are implemented, the population of this species is expected to be
𝑁 𝑡 = 3𝑡 3 + 2𝑡 2 − 10𝑡 + 600
(0 ≤ 𝑡 ≤ 10)
• where 𝑁 𝑡 denotes the population at the end of year 𝑡. Find the rate of growth of the
whale population when 𝑡 = 2 and 𝑡 = 6. How large will the whale population be 8 years
after the conservation measures are implemented?
Example 8
• An experimental rocket lifts off vertically, Its altitude (in feet) 𝑡 seconds into flight is given
by 𝑠 = 𝑓 𝑡 = −𝑡 3 + 96𝑡 2 + 5
𝑡≥0
• (a) Find an expression 𝑣 for the rocket’s velocity at any time 𝑡.
• (b) Compute the rocket’s velocity when 𝑡 = 0, 30, 50, 64, and 70. Interpret your results.
• (c) Using the results from the solution to part (b) and the observation that at the highest
point in its trajectory the rocket’s velocity is zero, find the maximum altitude attained by
the rocket.