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)?
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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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