1. Differentiation Formulas
2.3
Differentiation Formulas

2. Differentiation Formulas
Let’s start with the simplest of all functions, the constant function f (x) = c. The graph of this function is the horizontal line y = c, which has slope 0, so we must have f (x) = 0.
The graph of f (x) = c is the line y = c, so f (x) = 0.

3. Constant Rule
Using the formal definition of derivative:

4. Power Rule
For functions f (x) = xn, where n is a positive integer:

5. Proof by formal definition of derivative:
For n = 4 we find the derivative of f (x) = x4 as follows:
(x4) = 4x3

6. Practice: Find each derivative
(a) If f (x) = x6
(b) If y = x1000
(c) If y = t 4
(d) (r3)

7. Extended Power Rule

8. PROPERTIES AND RULES OF DERIVATIVES

9. Constant Multiple Rule

10. Sum Rule
the derivative of a sum of functions is the sum of the derivatives.

11. Difference Rule

12. Example: (x8 + 12x5 – 4x4 + 10x3 – 6x + 5)

13. Product Rule
Or:
the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

14. Quotient Rule
Or:
Or:
the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

15. Example:
Let Then

16. Use quotient rule? f(x) = f(x) = 3x + 2x –1 2
Don’t use the Quotient Rule every time you see a quotient.
Sometimes, when there is only ONE term in the quotient, it’s easier to rewrite the expression as a sum of power terms, then use the power rule.
Example:
f(x) =
We can use the quotient rule but it is much easier to perform the division first and write the function as:
f(x) = 3x + 2x –1 2

17. Examples:
(a) If y = , then
= –x –2 =
(b)

18. Practice: Differentiate the function f (t) = (a + bt).
Solution 1: Using the Product Rule, we have

19. Practice – Solution 2
cont’d
If we first use the laws of exponents to rewrite f (t), then we can proceed directly without using the Product Rule.

20. Normal line at a point:
The differentiation rules enable us to find tangent lines without having to resort to the definition of a derivative.
They also enable us to find normal lines.
The normal line to a curve C at point P is the line through P that is perpendicular to the tangent line at P.

21. Example:
Find equations of the tangent line and normal line to the curve
y = (1 + x2) at the point (1, ).
Solution: According to the Quotient Rule, we have

22. Example – Solution So the slope of the tangent line at (1, ) is
cont’d
So the slope of the tangent line at (1, ) is
We use the point-slope form to write an equation of the tangent line at (1, ):
y – = – (x – 1) or y =

23. Example – Solution
cont’d
The slope of the normal line at (1, ) is the negative reciprocal of , namely 4, so an equation is
y – = 4(x – 1) or y = 4x –
The curve and its tangent and normal lines are graphed in Figure 5.
Figure 5

24. Summary of Rules

25. Derivatives of Trig Functions
2.4
Derivatives of Trig Functions

28. Example 1
Differentiate y = x2 sin x.
Solution:
Using the Product Rule

29. Example 2
An object at the end of a vertical spring is stretched 4 cm beyond its rest position and released at time t = 0 (note that the downward direction is positive.)
Its position at time t is
s = f (t) = 4 cos t
Find the velocity and acceleration
at time t and use them to analyze
the motion of the object.

30. Example 2 – Solution
The velocity and acceleration are

31. Example 2 – Solution cont’d
The object oscillates from the lowest point (s = 4 cm) to the highest point (s = –4 cm). The period of the oscillation is 2, which is the period of cos t.

32. Example 2 – Solution cont’d
The speed is | v | = 4 | sin t |, which is greatest when | sin t | = 1, that is, when cos t = 0.
So the object moves fastest as it passes through its equilibrium position (s = 0). Its speed is 0 when sin t = 0, that is, at the high and low points.
The acceleration a = –4 cos t = when s = 0. It has greatest magnitude at the high and low points.