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OEF derivatives

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 cm^{2}/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 cm^{2}/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 cm^{2}, 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.

Rectangle I

We have a rectangle whose at a constant speed of centimeters per second, but whose stays constant at . At the moment when equals , what is the speed (in ) at which changes?

Rectangle II

We have a rectangle whose at a constant speed of centimeters per second, but whose stays constant at . At the moment when equals , what is the speed (in ) at which changes?

Rectangle III

We have a rectangle whose at a constant speed of centimeters per second, but whose stays constant at . At the moment when equals , what is the speed (in ) at which changes?

Rectangle IV

We have a rectangle whose at a constant speed of centimeters per second, but whose stays constant at . At the moment when equals , what is the speed (in ) at which changes?

Rectangle V

We have a rectangle whose at a constant speed of centimeters per second, but whose stays constant at . At the moment when equals , what is the speed (in ) at which changes?

Rectangle VI

We have a rectangle whose at a constant speed of centimeters per second, but whose stays constant at . At the moment when equals , what is the speed (in ) at which changes?

Right triangle

We have a right triangle as follows, where AB= , and AC at a constant speed of /s. At the moment when AC= , what is the speed at which BC changes (in /s)?

Tower

Somebody walks towards a tower at a constant speed of meters per second. If the height of the tower is meters, at which speed (in m/s) the distance between the man and the top of the tower decreases, when the distance between him and the foot of the tower is meters?

Trigonometric functions I

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

Trigonometric functions II

Trigonometric functions III

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

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Description: collection of exercises on derivatives of functions of one variable. interactive exercises, online calculators and plotters, mathematical recreation and games

Keywords: interactive mathematics, interactive math, server side interactivity, analysis, calculus, derivative, function, limit