Circle

We have a circle whose radius increases at a constant speed of $val6 centimeters per second. At moment time when the radius equals $val7 centimeters, what is the speed at which its area increases (in cm2/s)?

Circle II

We have a circle whose radius increases at a constant speed of $val6 centimeters per second. At moment time when its area equals $val7 square centimeters, what is the speed at which the area increases (in cm2/s)?

Circle III

We have a circle whose area increases at a constant speed of $val6 square centimeters per second. At the moment when the area equals $val7 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 $val6 square centimeters per second. At the moment when its radius equals $val7 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-3-2-10123
f(x) $val7$val8$val9 $val10$val11$val12 $val13
f '(x) $val14$val15$val16 $val17$val18$val19 $val20
g(x) $val21$val22$val23 $val24$val25$val26 $val27
g'(x) $val28$val29$val30 $val31$val32$val33 $val34

Let h(x) = f(g(x)). Compute the derivative h'($val35).


Composition II *

We have 3 differentiable functions f(x), g(x) and h(x), with values and derivatives shown in the following table.

x-3-2-10123
f(x) $val7$val8$val9 $val10$val11$val12 $val13
f '(x) $val14$val15$val16 $val17$val18$val19 $val20
g(x) $val21$val22$val23 $val24$val25$val26 $val27
g'(x) $val28$val29$val30 $val31$val32$val33 $val34
h(x) $val35$val36$val37 $val38$val39$val40 $val41
h'(x) $val42$val43$val44 $val45$val46$val47 $val48

Let s(x) = f(g(h(x))). Compute the derivative s'($val49).


Mixed composition

We have a differentiable function f(x), with values and derivatives shown in the following table.

x-2-1012
f(x) $val7$val8$val9 $val10$val11
f '(x) $val12$val13$val14 $val15$val16

Let g(x) = $val30, and let h(x) = g(f(x)). Compute the derivative h'($val17).


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-2-1012
f(x) $val7$val8$val9 $val10$val11
f '(x) $val12$val13$val14 $val15$val16
g(x) $val22$val23$val24 $val25$val26
g'(x) $val27$val28$val29 $val30$val31

Let h(x) = f(x)/g(x). Compute the derivative h'($val37).


Mixed division

We have a differentiable function f(x), with values and derivatives shown in the following table.

x-2-1012
f(x) $val7$val8$val9 $val10$val11
f '(x) $val12$val13$val14 $val15$val16

Let h(x) = $val30 / f(x). Compute the derivative h'($val17).


Hyperbolic functions I

Compute the derivative of the function f(x) = $val15.

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-2-1012
f(x) $val7$val8$val9 $val10$val11
f '(x) $val12$val13$val14 $val15$val16
g(x) $val22$val23$val24 $val25$val26
g'(x) $val27$val28$val29 $val30$val31

Let h(x) = f(x)g(x). Compute the derivative h'($val37).


Multiplication II

We have two differentiable functions f(x) and g(x), with values and derivatives shown in the following table.

x-2-1012
f(x) $val7$val8$val9 $val10$val11
f '(x) $val12$val13$val14 $val15$val16
f ''(x) $val17$val18$val19 $val20$val21
g(x) $val22$val23$val24 $val25$val26
g'(x) $val27$val28$val29 $val30$val31
g''(x) $val32$val33$val34 $val35$val36

Let h(x) = f(x)g(x). Compute the second derivative h''($val37).


Mixed multiplication

We have a differentiable function f(x), with values and derivatives shown in the following table.

x-2-1012
f(x) $val7$val8$val9 $val10$val11
f '(x) $val12$val13$val14 $val15$val16

Let h(x) = $val35 f(x). Compute the derivative h'($val22).


Virtual multiplication I

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

Polynomial I

Compute the derivative of the function f(x) = $val12, for x=$val10.

Polynomial II

Compute the derivative of the function .

Rational functions I


Rational functions II


Inverse derivative

Let $val6: $val8 -> $val8 be the function defined by

$val6(x) = $val20 .

Verify that $val6 is bijective, therefore we have an inverse function $val7(x) = $val6-1(x). Calculate the value of derivative $val7 '($val16) .

You must reply with a pricision of at least 4 significant digits.


Rectangle I

We have a rectangle whose $val19 $val7 at a constant speed of $val9 centimeters per second, but whose $val16 stays constant at $val17 $val18. At the moment when $val20 equals $val21 $val22, what is the speed (in $val25) at which $val26 changes?

Rectangle II

We have a rectangle whose $val19 $val7 at a constant speed of $val9 centimeters per second, but whose $val16 stays constant at $val17 $val18. At the moment when $val20 equals $val21 $val22, what is the speed (in $val26) at which $val27 changes?

Rectangle III

We have a rectangle whose $val19 $val7 at a constant speed of $val9 centimeters per second, but whose $val16 stays constant at $val17 $val18. At the moment when $val20 equals $val21 $val22, what is the speed (in $val26) at which $val27 changes?

Rectangle IV

We have a rectangle whose $val19 $val7 at a constant speed of $val9 centimeters per second, but whose $val16 stays constant at $val17 $val18. At the moment when $val20 equals $val21 $val22, what is the speed (in $val26) at which $val27 changes?

Rectangle V

We have a rectangle whose $val19 $val7 at a constant speed of $val9 centimeters per second, but whose $val16 stays constant at $val17 $val18. At the moment when $val20 equals $val21 $val22, what is the speed (in $val25) at which $val26 changes?

Rectangle VI

We have a rectangle whose $val19 $val7 at a constant speed of $val9 centimeters per second, but whose $val16 stays constant at $val17 $val18. At the moment when $val20 equals $val21 $val22, what is the speed (in $val25) at which $val26 changes?

Right triangle

We have a right triangle as follows, where AB=$val6 $val13, and AC $val12 at a constant speed of $val8 $val13/s. At the moment when AC=$val7 $val13, what is the speed at which BC changes (in $val13/s)?


Tower

Somebody walks towards a tower at a constant speed of $val6 meters per second. If the height of the tower is $val7 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 $val8 meters?

Trigonometric functions I

Compute the derivative of the function f(x) = $val15.

Trigonometric functions II


Trigonometric functions III

Compute the derivative of the function f(x) = $val16 at the point x=$val15.