Transcript Diapositive 1

CALCULUS I
Chapter II
Differentiation
Mr. Saâd BELKOUCH
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
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[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
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=
=
(combine terms)
= 32 x +16 h cancel common h terms
Thus, the derivative of f(x) = 16x2 is the function
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=32x
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Tangent’s slope & instantaneous rate of change
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Slope as a Derivative The slope of the tangent line to
the curve y = f(x) at the point (c,f(c))is
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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
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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?
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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
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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
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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
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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 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 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
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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:
=(
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Differentiate the product P(x) = (x - 1)(3x - 2) by a) Expanding P(x) b) The product rule.
a)
We have P(x) = 3
b)
By the product rule:
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- 5x + 2, so P'(x) = 6x - 5.
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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) =
=
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≠
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) = 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
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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
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Example: Find the fifth derivative of: f(x) = 4x3 + 5x2 + 6x – 1
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END OF CHAPTER II
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