The derivative of a function is a new function that describes the slope of the original function. Slope can be interpreted as the change in the dependent variable per unit change in the independent variable. In this case, the dependent variable is cost and the independent variable is the number of widgets produced. Therefore, the derivative of C(x) tells us the change in cost per change of one in the number of widgets produced.

To find a point on the line evaluate the function at the value of interest.  

To find the slope of the perpindicular line, the derivative of the function should be evaluated at the point of interest. 

The slope of a perpindicular line  at a given point is the negative reciprocal of the derivative at the point.

Thus the slope is 

.

Using the point slope form the equation is:

This limit can be evaluated using L'Hopital's rule, where the derivatives of the top and bottom are taken.

Thus:

To determine if a limit exists there are three properties it must have.

1) The limit from the left side of the point in question exists.

2) The limit from the right side of the point in question exists.

3) The limits from the left and right had side must be equal.

By factoring:

Therefore, the limit does not exist. 

The limit of a funtion to a certain x can be found by plugging x into the equation.

The limit can be evaluated by plugging the x into the limit expression. If the expression is undefined then we must use L'Hopital's rule. We can take the derivative of both the top and bottom of the fraction and take the limit of that. The derivative of the top is . The derivative of the bottom is .

To find where a function is increasing or decreasing, you must first identify the function's critical points. To do this, you find where the derivative of the function is greater than zero (for increasing), or less than zero (for decreasing). Convention is to include the point where the derivative equals zero in the interval.

Recall the following rule of differentiation to help solve this problem.

Power Rule: 

The derivative of the given function, by the power rule, is 

This derivative equals zero at . 

 when 

Therefore, the function is increasing when , or on the interval 

Knowing the values for and  in the limit as  approaches , we can plug them into the fraction and reduce to get a value.

We need the top function minus the bottom function in order to determine the area enclosed between these constraints. We also need to know the bounds on the integral.

We have two functions where  and  . Therefore, we need the first function minus the second. This is because the function zero is below the cubed function. Then, we are told that the  is a constraint.  That means that we are not allowed to integrate past it and that becomes our upper bound. Now we notice that the two functions with y meet at the origin, this gives us our lower bound of integration.

Recall the power rule for integration: 

We get:

Use the Power Rule to find the derivative 

.

Substitute  and get 

. 

The units of the derivative are .

The derivative of velocity, usually measured in meters per second (or units of length per units of time squared), is called accerelation.

The accerelation at  is .

Note: The Power Rule says that for a function , .