Derivative Rules
Suppose
f(x) = 13ln(\pi^2)e^4.
Compute f'(x).
f'(x) = 0.
Suppose
f(x) = x^4 - 13x^2 + 2.
Determine f''(x).
f''(x) = 12x^2 - 26.
Suppose that g(1) = 3, g'(1) = 1, h(1) = 2, and h'(1) = -1. If f(x) = g(x)h(x), compute f'(1).
f'(1) = -1.
Suppose that the position of an object, in meters, is given by
x(t) = t^3 + 4t + 1
at time t seconds. Compute the instantaneous rate of change of the position (or velocity) of the object at time t = 4 seconds.
52 meters/second.
Does the function
f(x) = 13x - 2
attain a maximum on (1,10)?
No
Suppose
f(x) = 15x^4-3x^3+7\sqrt{x} + 5/x.
Compute f'(x).
f'(x) = 60x^3 - 9x^2 + \frac{7}{2\sqrt{x}} - 5/x^2.
Determine the slope of the tangent line to the function
f(x) = 2x^2 - 4x + 7.
at x = 5.
16
Suppose that g(0) = 3, g'(0) = 2, g(1) = 4, and g'(1) = 5. If
f(x) = g(e^x)
compute f'(0).
f'(0) = 5.
The concentration, in parts per million, of nitrogen in a tank t seconds after the tank begins pressurization is given by
C(t) = \frac{3t^2}{4t^2+1}.
Compute the instantaneous rate of change of nitrogen concentration at time t = 2 seconds.
12/289 parts per million per second.
State the Extreme Value Theorem.
If f(x) is continuous on the closed interval [a,b], then f achieves a maximum and a minimum on [a,b].
Suppose
f(x) = (x^2 + 7)^3e^x
Compute f'(x).
f'(x) = 6x(x^2 + 7)^2e^x + (x^2 + 7)e^x.
Suppose
f(x) = 2\sqrt{x} + root(3)(x).
Compute f''(x).
f''(x) = -\frac{1}{2}x^{-3/2} - \frac{2}{9}x^{-5/3}.
Suppose that
f(x) = x^3 - 2
and that g(10) = 5 and g'(10) = 1. If h(x) = f(x)g(4x+2), compute h'(2).
h'(2) = 84.
The total revenue (in dollars) from selling x items is given by R(x) = 1100x − x^2/11 . Determine the average revenue function.
1100 - x/11
Determine the minimum value of
f(x) = x^3+3x^2-45x+2
on the interval [1,5].
-79
Suppose
f(x) = e^{root(5)(x^3+7x-2)}.
Compute f'(x).
\frac{1}{5}e^{root(5)(x^3+7x-2)(x^3+7x-2)^{-\frac{4}{5}}(3x^2+7).
If
f(x) = e^{2x},
then compute
f^{(10)}(x).
f^{(10)}(x) = 1024e^{2x}.
Suppose that g(5) = 12 and g'(5) = 10. If
f(x) = 4ln(g(x)),
compute f'(5).
f'(5) = \frac{10}{3}.
The cost function (in dollars) for producing x items is given by C(x) = 1100x + 11x - x^2/11, and the price per item is given by p = 1100 + 11/x. Determine the marginal profit.
11-{2x}/11
Determine the minimum value of
f(x) = \frac{3x^2+1}{x^3+1}
on the interval [-0.5, 2].
HINT: -3x^4-3x^2+6x=0 has roots at x=0 and x=1 .
1
Suppose
f(x) = \frac{14x^3}{4e^{35x^2+10x-3}}.
Compute f'(x).
f'(x) = \frac{168x^2e^{35x^2+10x-3}-56(70x+10)x^3e^{35x^2+10x-3}}{(4e^{35x^2+1x-3})^2
Determine the slope of the tangent line to the function
f(x) = 2x^3e^{2x}
at x = 2.
56e^4.
Suppose that
f(x) = root(4)(g(x))
and that the equation of the tangent line to g(x) at x = 2 is
y = -4(x - 2) + 16.
Compute f'(2).
f'(2) = -\frac{1}{8}.
The cost function (in dollars) for producing x items is given by C(x) = 1100x + 11x - x^2/11, and the price per item is given by p = 1100 + 11/x. Determine the marginal average profit.
1/11 -11/x^2
Determine the maximum value of
f(x) = {(-x^2+8x+3,if x<9),(5x-51,if x>=9):}
on the interval [0,10]
19