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 | -1 | 0 | 1 | 2 | 3 |
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 | -1 | 0 | 1 | 2 | 3 |
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 | -1 | 0 | 1 | 2 |
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 | -1 | 0 | 1 | 2 |
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 | -1 | 0 | 1 | 2 |
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 | -1 | 0 | 1 | 2 |
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 | -1 | 0 | 1 | 2 |
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 | -1 | 0 | 1 | 2 |
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.