Differential calculus (1)

Derivatives

100

What is the derivative of f(x) = (x^2 - 4) / x ?

1 - 4 / x^2

200

Given that f(x) = 3x + 2 + 1/x, find f"(-1).

-2

300

Differentiate y = {1 + (x^2 - 1)^3}^(1/3).

dy/dx = (2x (x^2 - 1)^2) / (1 + (x^2 - 1)^3)^(2/3)

400

Find dy/dx is y = (x + 4) / (x^2 - 1)^0.5

(2x^2 + 4x - 1) / (x^2 - 1)^0.5

500

Find the value of dy/dx at x = 0, given that y^3 - xy^2 - x^3 = 1.

1/3

Slopes, tangents, and normals

100

Find the gradient of the curve x^3 + y^3 = 4y^2 at the point (2, 2).

dy/dx = 3

Maximum and minimum

100

Find the greatest value of y = x^3 - 5x^2 + 7x on [0, 4].

y(max) = 12, when x = 1

200

A particle moves along a straight line so that its velocity v when it is s meters from a fixed point is given by v = s^2 + 3. Find an expression for its acceleration in terms of s.

a = 2s(s^2 + 3)

Related rates

100

A piece of paper is burning around the edges of a circular hole. After t seconds, the radius (r cm) of the whole is increasing at the rate of 0.5 cm/s. Find the rate at which the area of the whole is increasing when the radius is 5 cm.

15.7 cm^2/s

200

Water is being poured into a conical vessel at a rate of 10 cm^3/s. After t seconds, the volume V of the water in the vessel is given by V = (Pi/6)x^3, where x is the depth of the water in cm. Find, in terms of x, the rate at which the water is rising.

dx/dt = 20 / (Pi x^2)

Harder derivatives

100

Find dy/dx, if y^2 = tan2x + sec 2x.

dy/dx = y sec2x

200

Find the derivative of y = arcsin(3x - 1).

3 / (6x - 9x^2)^0.5

300

Find dy/dx, given that y = 2^x.

dy / dx = 2^x ln2