Begin by writing the expression for the area of a rectangle:

The rate of change of the area can be found by taking the derivative of the equation with respect to time:

Now, we're told two things:

The width of a rectangle increases at half the rate of its length:

The length of the rectangle is four times its width: 

Using this, rewrite the area equation in terms of width:

The rate of change of the area is  times the rate of change of the rate of change of the width.

To approach this problem, begin by defining the cylinder's volume and surface area in terms of its height and radius:

Rates of change can be found by deriving, then, with respect to time:

We're told two things:

The rate of change of a cylinder's radius is equal to the rate of change of its height:

The radius is twice the height: 

Using these properties, rewrite the rate equations:

The comparison between the volume and surface area can be found by taking the ratio of the two:

To approach this problem, begin by defining the cylinder's surface area in terms of its height and radius:

Rates of change can be found by deriving, then, with respect to time:

We're told two things:

The rate of change of a cylinder's radius is half the rate of change of its height: 

The radius is a fourth the height:

Using these properties, rewrite the rate equations:

To approach this problem, begin by defining the cylinder's volume and surface area in terms of its height and radius:

Rates of change can be found by deriving, then, with respect to time:

We're told two things:

The rate of change of a cylinder's radius is equal to twice the rate of change of its height: 

The radius is thrice the height: 

Using these properties, rewrite the rate equatios:

Begin by writing the equations for a cube's dimensions. Namely its volume and surface area in terms of the length of its sides:

The rates of change of these can be found by taking the derivative of each side of the equations with respect to time:

Now, solve for the length of the side of the cube to satisfy the problem condition,

Begin by writing the expression for the volume of a rectangular prism:

The rate of change of the volume can be found by taking the derivative of the equation with respect to time:

Now, we're given some information:

The width of a rectangular prism increases three times as fast as its length and twice as fast as its height:

The length, height, and width are equal: 

Using this, rewrite the volume equation in terms of width:

The rate of change of the volume is  times the rate of change of the rate of change of the width.

Using  ,

We have that:

 Which will give us the general derrivative of the function. Expanding, we have:

Simplifying:

Factoring out a d:

canceling out the 's

Applying the limit, we have:

Now, to find the instantaneous rate of change at  we evaluate the derrivative at :

, which is our answer.

We'll try to find the derivative:

:

This is a sum of three distinct terms, so we can apply the derivative to each term seperately:

We apply our knowledge of derivatives of the various functions:

 ;  ; 

So, we have:

Simplifying:

.

Evaluating this at , we get:

Equivalently:

  which is our answer.

We find the derivative with respect to .

As a sum of different functions, the derrivative can be applied to each of the different terms separately. 

Now, to simplify, we can move the coeficient out of the braces.

We use the facts

 ;  and 

to differentiate the function:

 giving us the answer.

Taking the derrivative

We can differentiate each term separately.

Factoring out the coeficients

We use the normal derivative rules:

Simplifying:

Evaluating at 

Simplifying

Which is our answer.