1. Chapter 2 Differentiation: Basic Concepts
In this Chapter, we will encounter some important concepts.
The Derivative
Product and Quotient Rules, Higher-Order Derivatives
The Chain Rule
Marginal Analysis, Implicit Differentiation.

2. Review: Techniques of Differentiation
Constant Rule: derivative of a constant is zero
Power Rule: for any real n,
Constant Multiple Rule: for constant c and a differentiable function f(x),
Sum Rule: when f(x) and g(x) are both differentiable,

3. Exercise

4. Solution

5. Exercise Find the equation of the line that is tangent to the graph of the function at the point (-1,-8).

6. Suggested Solution: Suppose the equation of the tangent line is m= at the point (-1,-8). So at the point (-1,-8), Hence, , and therefore -8=(-10)(-1)+b, gives b=2. The equation of the tangent line to this function at point (-1,-8) is

7. Review:
Relative rate of change:
Percentage rate of change:

8. Exercise 1.Find the relative rate of change of
with respect to x for the value x=1.
2.The gross annual earnings of a certain company were
thousand dollars t years after its formation in 2004.
At what rate were the gross annual earnings of the company growing with respect to time in 2008?
At what percentage rate were the gross annual earnings of the company growing with respect to time in 2008?

9. 1.Suggested Solution: The function is Its rate of change is
Its relative rate of change is
When x=1,

10. 2.Suggested Solution: (1)
, which is the rate that company’s gross annual earning changes (G.A.E) with respect to t years after 2004.
In 2008, t=4, the G.A.E will change at A’(4)=0.2*4+10=10.8 thousand dollars per year.
The percentage rate of that of the company’s G.A.E changes is expressed as So in year 2008, this value is
This means the company’s G.A.E increases 17.53% per year in the year 2008.

11. Review:
Product Rules: For function y=f(x) and z=g(x). If they are both differentiable at x, then the derivative of their product is
Quotient Rules: For function y=f(x) and z=g(x). If they are both differentiable at x, then the derivative of their quotient is
iff z=g(x) ≠ 0

12. Review: Example: Function S(t) Position
Second Order Derivative: is the derivative of the first derivative. Notation as follow:
Example:
Function S(t) Position
First Order Derivative S’(t)=V(t) Velocity
Second Order Derivative S”(t)=V’(t)=A(t) Acceleration

13. Compute the worker’s rate of production.
Exercise:
An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at 8:00am will have produced units t hours later.
Compute the worker’s rate of production.
At what rate is the worker’s rate of production changing with respect to time at 9:00am?

14. Suggested Solution
Worker’s rate of production is
(b) The rate that worker’s rate of production changes is
So at 9:00am, t=1

15. Review:
Chain Rule: For functions y=f(u), which is differentiable at u, and z=g(x), which is differentiable at x.
The composition function y=f(z)=f(g(x)) is differentiable at x, which is
!!! Pay Attention to the phrase “WITH RESPECT TO”

16. Exercise
Differentiate
Suggested Solution

17. Review:
Marginal Cost: If C(x) is the total cost of producing x units of a commodity.
Then the marginal cost of producing units is the derivative ,
which approximates the additional cost incurred when the level of production is increased by one unit, from to
Marginal Revenue :
Marginal Profit :

18. Review:
Approximation by Increment: If y=f(x) is differentiable at , and △x is a small change in x, then 
Approximation Percentage of change: if △x is a small change in x, then

19. Exercise
Use Incremental Approximation to approximate the value of Remember f(x+x) – f(x) = f  f’(x)x

20. Suggested solution Let f(x) = x80, x=1, x = 0.003 f’(x) = 80x79.
When x = 1, f’(x) = 80
f  80x = 0.24
f(x+x) – f(x) = 0.24.
=1+0.24=1.24

21. The ratio of Errors (output to input)
Suppose x is the correct (or precise) input
And x+x is the incorrect (or approximate) input.
Then x is called the error of the input.
and y = f(x +x) - f(x) is called the error of the output
If x is small, then y’  y/ x.
Or y = y’  x.

22. Example 23
During a medical procedure, the size of a roughly spherical tumor is estimated by measuring its diameter and using the
formula to compute its volume. If the diameter is measured as 2.5 cm with a maximum error of 2%, how accurate is the volume measurement?
Solution:
A sphere of radius R and diameter x=2R has volume
The error made in computing this volume using the diameter 2.5， while the actual diameter is 2.5+△x， is

23. Exercise
At a certain factory, the daily output is units, where K denotes the capital investment measured in units of $1,000. The current capital investment is $900,000. Estimate the effect that an additional capital investment of $800 will have on the daily output.

24. Suggested Solution
The current capital investment K₀=900 thousand dollars.
The increase in capital investment △K=0.8 thousand dollars.
To estimate the effect of this △K on the daily output:
This means an additional capital investment of 800 dollars would increase the daily output by 8 units.

25. Review: Differentials: Differentials of x is dx=△x (small increment)
If y=f(x) is a differentiable function of x, then the differential of y is

26. Review: Exercise: Differentiate implicitly with respect to x.
Implicit Differentiation:
Explicit Form: a function that can be written as y=f(x)
“y is solved, and given by an equation of x”
Implicit Form: cannot express y as a equation of x.
How to differentiate an implicit equation?
(1) Differentiate both side with respect to x
 you now have a function containing x and y and y’.
(2) Express y’ in terms of x and y.
Exercise:
Differentiate implicitly with respect to x.

27. Differentiate implicitly with respect to x.
Exercise:
Differentiate implicitly with respect to x.
Suggested Solution:

28. Example
Suppose the output at a certain factory is units, where x is the number of hours of skilled labor used and y is the number of hours of unskilled labor. The current labor force consists of 30 hours of skill labor and 20 hours of unskilled labor.
Question: Use calculus to estimate the change in unskilled labor y that should be made to offset a 1-hour increase in skilled labor x so that output will be maintained at its current level.

29. Solution:

30. What is the total cost at this time and at what rate is it changing?
Example
The manager of a company determines that when q hundred units of a particular commodity are produced, the cost of production is C thousand dollars, where When 1500 units are being produced, the level of production is increasing at the rate of 20 units per week.
What is the total cost at this time and at what rate is it changing?

31. Exercise
When the price of a certain commodity is p dollars per unit, the manufacturer is willing to supply x thousand units, where
How fast is the supply changing when the price is $9 per unit and is increasing at the rate of 20 cents per week?

32. Exercise
A lake is polluted by waste from a plant located on its shore. Ecologists determine that when the level of pollutant is x parts per million (ppm), there will be F fish of a certain species in the lake, where
When there are 4000 fish left in the lake, the pollution is increasing at the rate of 1.4ppm/year. At what rate is the fish population changing at this time?