Math 142 Lecture Notes for Section 4.1
Section 4.1 -
1
Derivatives of Powers, Exponents, and Sums
Definition 4.1.1 (Constant Function):
If c is a constant real number and f (x) = c, the derivative of f (x) is
Definition 4.1.2 (Power Rule):
If n is a real number, and f (x) = xn , the derivative of f (x) is
Definition 4.1.3 (Constant Multiple Property):
If k is a real number and g(x) a function, with f (x) = k · g(x), the derivative of f (x) is
Definition 4.1.4 (Sum and Difference Property):
If f (x) and g(x) are both differentiable functions, with h(x) = f (x) + g(x), then the derivative of h(x)
is
If f (x) and g(x) are both differentiable functions, with h(x) = f (x) − g(x), then the derivative of h(x)
is
Math 142 Lecture Notes for Section 4.1
Example 4.1.5:
Compute the derivative of the following:
(a) f (x) = 10
(b) f (x) = π
(c) f (x) =
√
5
9
(d) f (x) = x4
(e) f (x) = x1/2
(f) f (x) =
(g) f (x) =
√
3
x5
5
x2
2
Math 142 Lecture Notes for Section 4.1
(h) y = 3x2 + 7x + 9
(i) m(t) =
(j) k(x) =
(k) y =
√
5t − t1/2 + e
x3 + 4x2 − 17 + x
x
3x2 + x4
√
5 x
3
Math 142 Lecture Notes for Section 4.1
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Example 4.1.6:
If a skydiver jumps from a plane 35,000 in the air, its height above the ground (in feet) after t seconds
is given by s(t) = 35000 − 20t2 .
(a) Compute s0 (t) and interpret.
(b) Compute s(10) and interpret s0 (10).
Example 4.1.7:
If f (x) = 5x6 − 125x4 , where does the graph of the function have a horizontal tangent line?
Math 142 Lecture Notes for Section 4.1
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Example 4.1.8:
Suppose the total cost (in dollars) of producing x video games is given by
C(x) = 25x2 − 12x + 100.
(a) Find C(10) − C(9) and interpret.
(b) Find C 0 (5) and interpret.
Definition 4.1.9 (Marginal Business Functions):
Approximate change in the dependent variable (cost, revenue, profit) when the independent variable
(quantity of items produced/sold) is changed by a single unit.
• Marginal Cost Function
• Marginal Revenue Function
• Marginal Profit Function
Math 142 Lecture Notes for Section 4.1
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Example 4.1.10:
The total profit in dollars of producing x video games is given by
P (x) = x2 + 40x − 25000
(a) Find the exact profit realized from the sale of the 100th video game.
(b) Use the marginal profit to approximate the profit realized from the sale of the 100th video game.
Definition 4.1.11:
Derivatives of Exponential and Logarithmic Functions
• If f (x) = bx , then
• If f (x) = ex , then
• If f (x) = ln x, then
• If f (x) = logb x, then
Math 142 Lecture Notes for Section 4.1
Example 4.1.12:
Find the derivative of the following functions:
(a) f (x) = 2ex
(b) f (x) = 7(5x )
(c) f (x) = ln x + 7
(d) h(x) = 4x2 −
√
3
x + log7 x3 − 5x
(e) y = ln(x7 ) + 3(2)x
Example 4.1.13:
Find the equation of the line tangent to the graph of f (x) = ex + ln x at x = 1.
Suggested Homework Problems: 1-75 (odd), 79, 97
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