Derivatives
Related Rates
Chain Rule
Tangent Lines
Properties, rules, theorems, etc
100
Find the derivative of the following function and simplify. f(x) = 4x −3x + 2x −pi
f'(x)=(12x^2)-6x+2
100
A 20-ft ladder is leaning against a house when the base starts to slide away. By the time the base is 10 ft from the house, the base is moving at the rate of 6 ft/sec How fast is the top of the ladder sliding down the wall?
dy/dt = -3.46 ft/sec
100
f(x) = (8x^3 + 7)^4
f '(x) = 4(8x^3 + 7)^3 · (24x^2)
100
Find the equation of the tangent line at (1, 5) if f(x) = x + 4
f(x) = x + 4
100
Identify this rule f'(x)g(x)+f(x)g'(x)
Product Rule
200
Find the derivative of the following function and simplify. f(x)= [(x^2)/(3)]-[3/(X^2)]
f'(x)= 2/3x+[6/(x^3)]
200
You are pumping water out of a tank. How rapidly will the fluid level inside a vertical cylindrical tank, with radius 3 meters, drop if we pump the fluid out at the rate of 2 meters cubed per minute?
dh/dt = .07 m/min
200
h(t) = (t^8 – 9t^3 + 3t + 2)^10
h'(t) = 10(t^8 - 9t^3 + 3t + 2) · (8t^7 – 27t^2 + 3)
200
Find the equation of the tangent line at the point (-2, -7) if f(x) = x2 + 4x - 3
y = -7
200
Identify this theorem f'(c)=[f(b)-f(a)]/(b-a)
Mean Value Theorem
300
Find the derivative of the following function and simplify. f(x)=(x^2)/[(x^2)-2]
f'(x)= -4x/[(x^2)-2]^2
300
If you are filling a conical tank of water and the water runs into it at the rate of 9 ft3/min. The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep?
0.3183 ft/min
300
f(x) = sin(x^2)
f '(x) = cos(x^2) · 2x
300
Find the equation of the tangent line at the point (3, 25) if f(x) = x3 +1.33x2 -5x +1
y = 29.98 x – 64.94
300
If f is continuous on [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) =k
Intermediate Value Theorem
400
Find the derivative of the following function and simplify. f(x)=(e^x)/[(e^x)-1]
f'(x)=(-e^x)/[(e^x)-1]^2
400
A 13-ft ladder is leaning against a house when the base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft/sec and the height is moving at the rate of 3 ft/sec. At what rate is the area of the triangle (formed by the ladder, house and ground) changing?
da/dt = 30.5 square feet per second
400
h(x) = cos(3x)
h'(x) = –3sin(3x)
400
Find the equation of the tangent line at the point (1, 2) if f(x) = (x-1)(x2 + x-1)
y = -x +3
400
integral of( du/a^2+u^2)=?
(1/a)arctan(u/a) +C
500
Find the derivative of the following function and simplify. f(x)=2x/(x-1)
f'(x)=-2/(x-1)^2
500
A 25-ft ladder is leaning against a house when the base starts to slide away. By the time the base is 20 ft from the house, the base is moving at the rate of 5 ft/sec. At what rate is the angle between the ladder and the house changing?
0.33 radians/second
500
s(u) = cos5(u)
s'(u) = –5cos^4(u)sin(u)
500
Find the equation of the tangent line at (5, 12/13) if f(t) = (t2-1)/(t2+1)
f(t) = 5/169 * t + 131/169
500
f(x)dx=F(b)-F(a)
Fundamental theorem
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