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