# 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 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 and , with values and derivatives shown in the following table.
x-3-2-10123
Let be defined by . Compute the derivative .

### Composition II *

We have 3 differentiable functions , and , with values and derivatives shown in the following table.
x-3-2-10123
Let the function defined by . Compute the derivative .

### Mixed composition

We have a differentiable function , with values and derivatives shown in the following table.
x-2-1012
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.
x-2-1012
Let defined by . Compute the derivative .

### Mixed division

We have a differentiable function , with values and derivatives shown in the following table.
x-2-1012
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.
x-2-1012
Let . Compute the derivative .

### Multiplication II

We have two differentiable functions and , with values and derivatives shown in the following table.
x-2-1012
Let . Compute the second derivative .

### Mixed multiplication

We have a differentiable function , with values and derivatives shown in the following table.
-2-1012
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 . The most recent version

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