#
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. 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. 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. 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=.
Other exercises on:
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