Basic Derivative Rules
d/dx (-4x-3)?
12x-4
d/dx (5xe2x)
10xe2x + 5e2x
Find the marginal average 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 constant rule?
d/dx (k (f(x))) = k (f'(x))
d/dx (8ex)?
8ex
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 (5x(6ex))?
30xex + 30ex
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.
f(x) = (x2 + 1); g(x) = ex - 1
f[g(x)] = (ex - 1)2 + 1
g[f(x)] = e(x^2)
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 + 51/2)?
15x2 - 14x - 9
d/dt ((5t2 - 7t) / (3t + 1)3)?
((3t + 1)3(10t - 7) - (5t2 - 7t)(9)(3t + 1)2) / (3t + 1)2
If a sum of $1000 is deposited into an account that pays r% interest compounded continuously, the balance after 12 years is given by
A = 1000e12r/100
Find and interpret dA/dr when r = 5.
(12/100) (1000) e12r/100
A'(5) = 218.65
When the rate increases from 5%, the account will increase by $218.65.
Johana sells silk-screen T-shirts at community festivals and craft fairs. Her marginal cost to produce one T-shirt is $3.50. Her total cost to produce 60 T-shirts is $300, and she sells them for $9.
a. Find the linear cost function for Johana's T-shirt production.
b. How many T-shirts must she sell to break even?
c. How many T-shirts must she produce and sell to make a profit of $500?
a. C(x) = 35x + 90
b. 17 T-shirts
c. 108 T-shirts
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
d/dx (e2x ln(xex + 1)) (Worth 700 Points)
e2x(xex + ex / xex + 1) + 2x2x(ln(xex + 1))
Find 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.
A company must pay a $307,000 settlement in 3 years.
a. What amount must be deposited now at 6% compounded semiannually to have enough money for the settlement?
b. How much interest will be earned?
c. Suppose the company can only deposit $200,000 now. How much more will be needed in 3 years?
a. $257,107.67
b. $49,892.33
c. $68,189.54
R(q) = q D(q)
Plug in the demand function and find the derivative