1. CALCULUS I
Chapter II
Differentiation
Mr. Saâd BELKOUCH

2. The derivative
Techniques of differentiation
Product and quotient rules, high-order derivatives

3. Section 1: The derivative
Derivatives are all about change, they show how fast something is changing (also called rate of change) at any point
Studying change is a procedure called differentiation
Examples of rate of change are: velocity, acceleration, production rate…etc
The derivative tell us how to approximate a graph, near some base point, by a straight line. This is what we call the tangent

4. Relationship between rate of change and slope

5. Derivative of a function
The derivative of the function f(x) with respect to x is the function f’(x) given by
[read f’(x) as “f prime of x”].The process of computing the derivative is called differentiation , and we say that f(x) is differentiable at x = c if f’( c) exists; that is ;if the limit that defines f’(x) exists when x=c.

6. Example 2.1 Find the derivative of the function f(x) = 16x2.
The difference quotient for f(x) is =
= (combine terms)
= 32 x +16 h cancel common h terms
Thus, the derivative of f(x) = 16x2 is the function
=32x

7. Tangent’s slope & instantaneous rate of change
Slope as a Derivative The slope of the tangent line to the curve y = f(x) at the point (c,f(c))is
Instantaneous Rate of Change as a Derivative The rate of change of f(x) with respect to x when x=c is given by f’(c ) .

8. Example 2.2
First compute the derivative of f(x) = x3 and then use it to find the slope of the tangent line to the curve y = x3 at the point where x = -1. What is the equation of the tangent line at this point?
According to the definition of the derivative =
Thus, the slope of the tangent line to the curve y = x3 at the point where x = -1 is f'(-1) = 3(-1)2 = 3
To find an equation for the tangent line, we also need the y coordinate of the point of tangency; namely, y = (-1)3 = -1.

9. Example 2.2 (cont.)
By applying the point-slope formula, we obtain the equation: y – (-1) =3 [x – (-1)] thus: y = 3 x+2

10. Sign of a derivative Significance of the Sign of the Derivative f’(x).
If the function f is differentiable at x = c ,then:
f is increasing at x =c if f’( c ) >0
f is decreasing at x =c if f ( c ) <0

11. Derivative notation
The derivative f'(x) of Y = f(x) is sometimes written
read as "dee y, dee x" or "dee f, dee x“
In this notation, the value of the derivative at x = c [that is, f ‘(c)] is written as
Continuity of a Differentiable Function
If the function f(x) is differentiable at x = c, then it is also continuous at x=c.

12. Section 2: Techniques of Differentiation
The constant Rule: For any constant c, (c) =0
that is ,the derivate of a constant is zero.
Example:
The Power Rule: For any real number n,
In words, to find the derivative of xn, reduce the exponent n of x by 1 and multiply your new power of x by the original exponent.
Examples:
The derivative of y =
Recall that so the derivative of y = is:
= =

13. The Constant Multiple Rule
If c is a constant and f(x) is differentiable, then so is cf(x) and
[cf(x)] = c
that is, the derivative of a multiple is the multiple of the derivative.

14. The Sum Rule
If f(x) and g(x) are differentiable, then so is the sum S(x) = f(x) + g(x) and S'(x) = f'(x) + g'(x);
that is, [f(x)+g(x)] = [g(x)]
In words, the derivative of a sum is the sum of the separate derivatives.
Example:

15. Section 3: Product and Quotient Rules; Higher-Order Derivatives
The product Rule: If f(x) and g(x) are differentiable at x, then so is their product P(x) = f(x) g(x) and:
 or equivalently,
In words ,the derivative of the product fg is f times the derivative of g plus g times the derivative of f.
Examples:
= (
Differentiate the product P(x) = (x - 1)(3x - 2) by a) Expanding P(x) b) The product rule.
We have P(x) = x + 2, so P'(x) = 6x - 5.
By the product rule:

16. The Quotient Rule: If f(x) and g(x) are differentiable functions ,then so is the quotient Q(x) = f(x)/g(x) and:
or equivalently: (
Recall that: ; but that ≠
Example: Differentiate the quotient Q(x) = by using the quotient rule. =

17. The Second Derivative
The second derivative of a function is the derivative of its derivative.
If y = f(x), the second derivative is denoted by or f’’(x)
The second derivative gives the rate of change of the rate of change of the original function.
Example: Find the second derivative of the function f(x) = 5x4 - 3x2 - 3x + 7.
Compute the first derivative
f ’(x) = 20 x3 - 6x - 3
then differentiate again to get
f ’’(x) = 60x2 - 6

18. High-Order Derivative
For any positive integer n, the nth derivative of a function is obtained from the function by differentiating successively n times. If the original function is y = f(x), the nth derivative is denoted by
Example: Find the fifth derivative of: f(x) = 4x3 + 5x2 + 6x – 1

19. END OF CHAPTER II