Basic Derivatives
What is the derivative of y = 3x2 ?
y' = 6x
Find y' if y = (3x - 9)4
12(3x - 9)3
Build a rectangular pen with three parallel partitions using 500 feet of fencing. What dimensions will maximize the total area of the pen ?
x=50 ft. and y=125 ft.
The equation of a curve is y = 6/(5-2x). A point moves along the curve in such a way that the rate of increase of y has a constant value of 0.02 unites per second. Find the rate of increase of x when x=1.
0.015
Find the derivative of f(x) = 5
0
Find the derivative of f(x) = sqrt(5x -2x2)
(5-4x)/2sqrt(5x-2x2)
An open rectangular box with square base is to be made from 48 ft.2 of material. What dimensions will result in a box with the largest possible volume ?
x=4 ft. and y=2 ft.
A watermelon is assumed to be spherical in shape while it is growing. Its mass Mkg, and radius r cm, are related by the formula M=kr3, where k is a constant. It is also assumed that the radius is increasing at a constant rate of 0.1 cm per day. On a particular day the radius is 10cm and the mass is 3.2kg. Find the value of k and the rate at which the mass is increasing on this day.
0.096
Find the derivative of f(x) = 6x3 + x - 2
18x2 + 1
Find y' if y = (9x2 + 4)1/3
(6x)/(9x2 + 4)2/3
A container in the shape of a right circular cylinder with no top has surface area 3 ft.2 What height h and base radius r will maximize the volume of the cylinder ?
r=1 ft. and h=1 ft.
The radius of a sphere is increasing at a rate of 2 meters per second. At what rate is the volume increasing when the radius is equal to 4 meters?
128Π cubic meters per second.
Find the derivative of f(x) = (12x3-5x)/(3x4) at x = 2
-0.6875 or -11/16
f(x) = 2/(3-4x), find f'(3).
8/81
A solid cylinder with radius x cm has a volume of 1000cm3. What is the minimum total surface area?
553.6 cm2
The volume of a spherical balloon is increasing at a constant rate of 50cm3 per second. Find the rate of increase of the radius when the radius is 10cm.
0.0398
Find the coordinates of the points on the curve
y=6 + 9x - 3x2 - x3 where the gradient is 9.
(0,6) and (-2,-16)
Given that f(x) = 4/sqrt(3x-2), find f"(2).
27/32
The horizontal base of a solid prism is an equilateral triangle of side x cm. The sides of the prism are vertical. The height of the prism is h cm and the volume of the prism is 2000 cm3 . What is the value of x that will minimize the surface area?
x=20cm
A rectangular block of ice has dimensions x by x by 3x. The ice is melting in such a way that the surface area is decreasing at a constant rate of 0.14cm2 per second. Calculate the rate of decrease of x at the instant when x = 2.
-0.0025