CALCULUS I Chapter II Differentiation Mr. Saâd BELKOUCH

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

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 3

Relationship between rate of change and slope 4

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. 5

Example 2.1  Find the derivative of the function f(x) = 16x 2.  The difference quotient for f(x) is = = (combine terms) = 32 x +16 h cancel common h terms  Thus, the derivative of f(x) = 16x 2 is the function =32x 6

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 ). 7

Example 2.2  First compute the derivative of f(x) = x 3 and then use it to find the slope of the tangent line to the curve y = x 3 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 = x 3 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. 8

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

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 10

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. 11

Section 2: Techniques of Differentiation The constant Rule: For any constant c, (c) =1 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 x n, 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: = = 12

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. 13

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: 14

Section 3: Product and Quotient Rules; Higher-Order Derivatives 15  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. a) We have P(x) = 3 - 5x + 2, so P'(x) = 6x - 5. b) 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. =

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) = 5x 4 - 3x 2 - 3x + 7.  Compute the first derivative f ’(x) = 20 x 3 - 6x - 3 then differentiate again to get f ’’(x) = 60x

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 18

END OF CHAPTER II 19