1. § 1.7
More About Derivatives

2. Section Outline Differentiating Various Independent Variables
Computing Second Derivatives
Second Derivatives Evaluated at a Point
Marginal Cost
Marginal Revenue
Marginal Profit

3. Differentiating Various Independent Variables
EXAMPLE
Find the first derivative.
SOLUTION
We first note that the independent variable is t and the dependent variable is T. This is significant inasmuch as they are considered to be two totally different variables, just as x and y are different from each other. We now proceed to differentiate the function.
This is the given function.
We begin to differentiate.
Use the Sum Rule.
Use the General Power Rule.

4. Differentiating Various Independent Variables
CONTINUED
Finish differentiating.
Simplify.
Simplify.

5. Second Derivatives EXAMPLE SOLUTION
Find the first and second derivatives.
SOLUTION
This is the given function.
This is the first derivative.
This is the second derivative.

6. Second Derivatives Evaluated at a Point
EXAMPLE
Compute the following.
SOLUTION
Compute the first derivative.
Compute the second derivative.
Evaluate the second derivative at x = 2.

7. Marginal Cost

8. Marginal Cost EXAMPLE SOLUTION
Let C(x) be the cost (in dollars) of manufacturing x bicycles per day in a certain factory. Assume C(50) = 5000 and Estimate the cost of manufacturing 51 bicycles per day.
SOLUTION
We will first use the additional cost formula for manufacturing 1 more bicycle per day beyond the cost of producing 50 bicycles per day. We already know it costs $5000 to produce 50 bicycles per day since C(50) = So we wish to determine how much more, beyond that $5000, it costs to produce 51 bicycles.
Therefore, we estimate the cost of manufacturing 51 bicycles to be $5045.

9. Marginal Revenue & Marginal Profit

10. Marginal Revenue EXAMPLE SOLUTION
Suppose the revenue from producing (and selling) x units of a product is given by R(x) = 3x – 0.01x2 dollars.
(a) Find the marginal revenue at a production level of 20.
(b) Find the production levels where the revenue is $200.
SOLUTION
(a) Since we are looking for the marginal revenue at a production level of 20, and we have an equation for R(x), we will simply find
This is the given revenue function.
This is the marginal revenue function.
Evaluate the marginal revenue function at x = 20.
Therefore, the marginal revenue at a production level of 20 is 2.6.

11. Marginal Revenue CONTINUED
(b) To find the production levels where the revenue is $200 we need to use the revenue function and replace revenue, R(x), with 200 and then solve for x.
This is the given revenue function.
Replace R(x) with 200.
Get everything on the left side of the equation.
Multiply everything by 100.
Factor.
Solve for x.
Therefore, the production levels for which revenue is $200 are x = 100 and x = 200 units produced.