Exercises

1. Calculate the derivative of each of the following functions.

   `x^(5//2)`	`x^(5//2)+x^2`	`(x^2+3)^(5//2)`
   `x^(-5//2)`	`sqrt(x^2+3)`	`(ln x)^2`

2. Differentiate each of the following functions.

   `x^(4//3) ln x`	`x(p^2+x^2)^(-1//2)`	`x^(-2//3) ln x`
   `(ln x+x)^(-1//3)`	`(1-sqrt(x)) ln x`	`(1+x^2)^(1//3)`

3. A rectangular computer chip is made of ceramic material with circuitry placed in an interior rectangle. Two parallel edges of the circuitry are each 1 millimeter from the corresponding edge of the chip, and the other two edges are each 2 millimeters from the corresponding edges of the chip. Find the length and width of the rectangular chip of area 200 square millimeters that maximizes the area of the circuitry it can accommodate.
4. In a recent (fictional) study, scientists formulated the following equation to represent the amount `i` of information retained from one day of studying:

   `i=k s c^(3//2)`,

   where `s` is the number of hours slept, `c` is the number of hours spent cramming, and `k` is a proportionality constant. Assuming that the average freshman prepares for finals by cramming every hour that she or he is not sleeping, how much should a student sleep in the course of a day to maximize the knowledge retained?

5. A truck traveling on a flat interstate highway at a constant rate of 60 mph gets 3.7 miles to the gallon. Fuel costs `\$`2.65 per gallon. For each mile per hour increase in speed, the truck loses a tenth of a mile per gallon in its mileage. Drivers get `\$`23.50 per hour in wages, and fixed costs for running the truck amount to `\$`17.33 per hour. What constant speed should a dispatcher require on a straight run through 260 miles of Kansas interstate to minimize the total cost of operating the truck?
   
   This exercise is adapted from Calculus Problems for a New Century, edited by Robert Fraga, MAA Notes No. 28, 1993.
6. Let `f` be the function defined by `ftext[(]xtext[)]=x^x`.
   1. Find a formula for `f'text[(]xtext[)]`. (Hint: `y=x^x` satisfies the equation `ln y=x ln x`.)
   2. Find the smallest value of `x^x`.