CALCULUS I

Chapter IIDifferentiation

Mr. Saâd BELKOUCH

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

derivatives

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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

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Relationship between rate of change and slope

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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.

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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

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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 ) .

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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.

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Example 2.2 (cont.)By applying the point-slope formula, we obtain the equation: y – (-1) =3

[x – (-1)]

thus: y = 3 x+2

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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

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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.

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Section 2: Techniques of DifferentiationThe 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 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:

= =

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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.

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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:

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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.

a) We have P(x) = 3 - 5x + 2, so P'(x) = 6x - 5.

b) By the product rule:

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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.

=

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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

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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

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END OF CHAPTER II