Basic Derivative Rules
d/dx (-4x-3)?
12x-4
d/dx (15x2+1)3
3(15x2+1)2(30x) or 90x(15x2+1)2
Find the marginal cost function:
C(x) = (x2 + 3)3
3(x2 + 3)2 (2x)
Find the value of x:
6x - x2 = x + 6
x = 2, 3
What is the power rule?
d/dx (c (f(x))) = c (f'(x))
d/dx (8)?
0
d/dt ((2t7 - 5)1/2)?
(1/2)(2t7 - 5)-1/2(14t6)
The sales of a company are related to its expenditures on research by
S(x) = 1000 + 60(x1/2) + 12x
where S(x) gives sales in millions when x thousand dollars is spent on research.
Find dS/dx. As the amount spent on research, what happens to sales?
30x-1/2 + 12
As the amount spent on research increased, sales approach 12,000,000.
The sales of a small company were $27,000 in its second year of operation and $63,000 in its fifth year. Let y represent sales in the xth year of operation.
Find the slope of the sales line, and give an equation for the line in the form y = mx + b.
y = 12000x + 3000
What's the formula for the product rule? f(x)g(x)
f'(x)g(x) + f(x)g'(x)
d/dx (x2+2x+1)?
2x+2
d/dt (t2(t2 + 1)5/2)
2t(t2 + 1)5/2 + t2(5/2(t2 + 1)3/2(2t))
The number (in billions) of pieces of mail handled by the U.S. Post Office each year from 2010 through 2019 can be approximated by
P(t) = -0.003584t4
where t is the number of years since 2010. Find and interpret the rate of change in the volume of mail for 2018.
0.014336t3
P'(8) = 7.340032
The volume of mail from 2018-2019 will increase by 7.34 billion.
Given (ex - 1)2 + 1, what is the inside function and what is the outside function?
outside = (x2 + 1); inside = ex - 1
What's the formula for the quotient rule? (f(x) / g(x))
(g(x)f'(x) - f(x)g'(x)) / (g(x))2
d/dx (5x3 - 7x2 - 9x + 5)?
15x2 - 14x - 9
d/dt ((5t2 - 7t) / (3t + 1)3)?
((3t + 1)3(10t - 7) - (5t2 - 7t)(9)(3t + 1)2) / (3t + 1)2
Find the marginal profit of a company with P(x) = x2+3x-5.
Find and interpret the marginal profit P'(20)
P'(x)=2x+3
P'(20)=43.
If one additional unit (the 21st unit) is made and sold, then the profit is expected to increase by about $43.
Johana sells silk-screen T-shirts at community festivals and craft fairs. Her total cost to produce x shirts is C(x) = 5x2 - 9, and she sells them for $19.
a. Find the marginal cost function for Johana's T-shirt production.
b. How many T-shirts must she sell to break even?
a. C ' (x) = 10x
b. 4.2 T-shirts so technically, 5.
What's the chain rule?
dy/dx = dy/du du/dx
or
d/dx f(g(x)) = f'(g(x)) g'(x)
d/dx (3x / (4x + 7))?
21 / (4x + 7)2
Find the second derivative of 1/(x+1)2
f ' (x) = -2(x+1)-3
f " (x) = 6(x+1)-4Find an interpret the marginal profit when 12 units are sold:
P(x) = x2/(2x+1)
(2x + 1)(2x) - (x2)(2) / (2x + 1)2
The profit will increase by $0.50 when producing 13 units.
How do we find the average cost function?
We take the cost function and divide by x.
Given a demand function of p(x) = 3x-4,
a. What is the revenue function?
b. What is the marginal revenue?
a. R(x) = xp(x) = 3x2-4x
b. R ' (x) = 6x-4