#
OEF Derivatives
--- Introduction ---

This module actually contains 35 exercises on derivatives of real
functions of one variable.

### Arc and Arg

Establish the correspondence between the fucntions and their derivatives in the following table.

### Circle

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

### Circle II

We have a circle whose radius increases at a constant speed of centimeters per second. At the moment when its area equals square centimeters, what is the speed at which the area increases (in
/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
and
, with values and derivatives shown in the following table. Let
be defined by
. Compute the derivative
.

### Composition II *

We have 3 differentiable functions
,
and
, with values and derivatives shown in the following table. Let
the function defined by
. Compute the derivative
.

### Mixed composition

We have a differentiable function
, with values and derivatives shown in the following table. Let
, and let
defined by
. Compute the derivative
.

### 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
and
, with values and derivatives shown in the following table. Let
defined by
. Compute the derivative
.

### Mixed division

We have a differentiable function
, with values and derivatives shown in the following table. Let
defined by
. Compute the derivative
.

### Hyperbolic functions I

Compute the derivative of the function
defined by
.

### Hyperbolic functions II

Compute the derivative of the function
defined by
.

### Multiplication I

We have two differentiable functions
and
, with values and derivatives shown in the following table. Let
. Compute the derivative
.

### Multiplication II

We have two differentiable functions
and
, with values and derivatives shown in the following table. Let
. Compute the second derivative
.

### Mixed multiplication

We have a differentiable function
, with values and derivatives shown in the following table. Let
defined by
. Compute the derivative
.

### Virtual multiplication I

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

### Polynomial I

Compute the derivative of the function
defined by
, for
.

### Polynomial II

Compute the derivative of the function
defined by
.

### Rational functions I

Compute the derivative of the function

### Rational functions II

Compute the derivative of the function

### Inverse derivative

Let
be the function defined by
.

Verify that
is bijective, therefore we have an inverse function
. Calculate the value of its derivative
at
. You must reply with a precision 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)?

### Sign of a number

Construct a study of the sign of
by choosing four of the sentences given below.

### Tower

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

### Trigonometric functions I

Compute the derivative of the function
defined by
.

### Trigonometric functions II

Compute the derivative of the function
.

### Trigonometric functions III

Compute the derivative of the function
defined by
at the point
.
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