OEF derivatives --- Introduction ---

This module actually contains 33 exercises on derivatives of real functions of one variable.

Circle

We have a circle whose radius increases at a constant speed of centimeters per second. At moment time when the radius equals centimeters, what is the speed at which its area increases (in cm²/s)?

Circle II

We have a circle whose radius increases at a constant speed of centimeters per second. At moment time when its area equals square centimeters, what is the speed at which the area increases (in cm²/s)?

Circle III

We have a circle whose area increases at a constant speed of square centimeters per second. At the moment when the area equals cm², what is the speed at which its radius increases (in cm/s)?

Circle IV

We have a circle whose area increases at a constant speed of square centimeters per second. At the moment when its radius equals cm, what is the speed at which the radius increases (in cm/s)?

Composition I

We have two differentiable functions f(x) and g(x), with values and derivatives shown in the following table.

x	-3	-2	-1	0	1	2	3
f(x)
f '(x)
g(x)
g'(x)

Let h(x) = f(g(x)). Compute the derivative h'().

Composition II *

We have 3 differentiable functions f(x), g(x) and h(x), with values and derivatives shown in the following table.

x	-3	-2	-1	0	1	2	3
f(x)
f '(x)
g(x)
g'(x)
h(x)
h'(x)

Let s(x) = f(g(h(x))). Compute the derivative s'().

Mixed composition

We have a differentiable function f(x), with values and derivatives shown in the following table.

x	-2	-1	0	1	2
f(x)
f '(x)

Let g(x) = , and let h(x) = g(f(x)). Compute the derivative h'().

Virtual chain Ia

Let be a differentiable function, with derivative . Compute the derivative of .

Virtual chain Ib

Let be a differentiable function, with derivative . Compute the derivative of .

Division I

We have two differentiable functions f(x) and g(x), with values and derivatives shown in the following table.

x	-2	-1	0	1	2
f(x)
f '(x)
g(x)
g'(x)

Let h(x) = f(x)/g(x). Compute the derivative h'().

Mixed division

We have a differentiable function f(x), with values and derivatives shown in the following table.

x	-2	-1	0	1	2
f(x)
f '(x)

Let h(x) = / f(x). Compute the derivative h'().

Hyperbolic functions I

Compute the derivative of the function f(x) = .

Hyperbolic functions II

Multiplication I

We have two differentiable functions f(x) and g(x), with values and derivatives shown in the following table.

x	-2	-1	0	1	2
f(x)
f '(x)
g(x)
g'(x)

Let h(x) = f(x)g(x). Compute the derivative h'().

Multiplication II

We have two differentiable functions f(x) and g(x), with values and derivatives shown in the following table.

x	-2	-1	0	1	2
f(x)
f '(x)
f ''(x)
g(x)
g'(x)
g''(x)

Let h(x) = f(x)g(x). Compute the second derivative h''().

Mixed multiplication

We have a differentiable function f(x), with values and derivatives shown in the following table.

x	-2	-1	0	1	2
f(x)
f '(x)

Let h(x) = f(x). Compute the derivative h'().

Virtual multiplication I

Let be a differentiable function, with derivative . Compute the derivative of .

Polynomial I

Compute the derivative of the function f(x) = , for x=.

Polynomial II

Compute the derivative of the function .

Rational functions I

Rational functions II

Inverse derivative

Let : -> be the function defined by

(x) = .

Verify that is bijective, therefore we have an inverse function (x) = ^-1(x). Calculate the value of derivative '() .

You must reply with a pricision of at least 4 significant digits.