7_EMA156_7S_Differentiation..........pdf

cput calculus slides

1. Engineering Mathematics 1
(EMA156/7S)
Differentiation
H. Mataifa
Department of Electronic, Electrical and
Computer Engineering
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2. Outline
•Multiple and sum rules
•Product and quotient rules
•Chain rule
•Differentiation of f(y)
•Implicit differentiation
•Logarithmic differentiation
•Higher derivatives
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3. Multiple and sum rules
•Multiple rule for differentiation: the derivative of the product of a constant and a function equals the
constant times the derivative of the function


•Example 1: use the multiple rule and determine the derivatives of the following functions:



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4. Multiple and sum rules
•Sum rule for differentiation: the derivative of a sum equals the sum of the derivatives. Thus, if f and g
are functions of x, then:


•Example 2: apply the sum rule of differentiation to the following functions:



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5. Product and quotient rules
•Let f and g be functions of x and denote their derivatives by f’ and g’ respectively
•Product rule for differentiation: states that:


•Quotient rule for differentiation: states that:



•Product rule in words: the derivative of the product of two functions equals the first function times the
derivative of the second function, plus the second function times the derivative of the first function
•Quotient rule in words: the derivative of a quotient equals the denominator function times the
derivative of the numerator function, minus the numerator times the derivative of the denominator, all
divided by the square of the denominator
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6. Product and quotient rules
•General product rule for differentiation: states that the derivative of a product of n factors equals the
sum of the n products obtained by multiplying the derivative of each factor by all the other factors.
Thus, for example, if p, q r and s are all functions of x, then:


•Example 3: apply the product and quotient rules to determine the derivatives of each of the following
functions and simplify:

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7. Chain rule
•If y=f(u) and u=g(x), then y=f(g(x)), and so y is a function of x, and we say that y is a composite
function of x. To differentiate a composite function, we use the chain rule, stated as:


•Thus, whenever we have y as a composite function of x, it is often convenient to express y as a
function of u, which is in turn a function of x, then apply the chain rule. For example, in:
y=sin3x, let u=3x, then y=sinu
y=√lnx, let u=lnx, then y=√u
y=ln√x, let u=√x, then y=lnu
y=e
tanx
, let u=tanx, then y=e
u
•In the above expressions, u is referred to as the argument of y; in turn, x is the argument of u; so to
differentiate y with respect to x, we first differentiate it with respect to its argument u, then multiply the
result by the derivative of u with respect to x
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8. Chain rule
•Example 4: apply the chain rule to determine the derivatives of each of the following functions:




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9. Differentiation of f(y)
•Suppose y=f(x) and we want to determine d/dx(y
3
) (i.e. differentiate y
3
with respect to x)?
•We can apply the chain rule. Let z=f(y)=y
3
. Then:


•Example 5: if y is a function of x, determine the following derivatives (apply the chain rule):

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10. Implicit differentiation
•Consider the following expressions:



•For each of the expressions above, it is not possible to express them in the form y=f(x). These
expressions are said to be in the implicit form (because it’s not possible to express y explicitly in terms
of x).
•To differentiate such expressions, we apply what we call implicit differentiation, as we show in the
following examples
•Example 6: determine dy/dx if:

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11. Differentiation of ln(f(x))
•If f(x) is a product or a quotient or a power, then the differentiation of ln(f(x)) can be simplified
considerably by application of the following logarithmic laws:




•Example 7: determine dy/dx if:

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12. Logarithmic differentiation
•If y=f(x) involves a product and/or a quotient and/or a power, it may sometimes be more convenient to
apply the log function and then determining the derivative using the resulting expression (i.e. lny=lnf(x))
by implicit differentiation:

•Example 8: determine the derivative of each of the following functions:

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13. Higher derivatives
•If y=f(x), then dy/dx is the rate at which y is increasing with respect to x (also referred to as the gradient
of y=f(x)). We can further determine the rate which dy/dx changes with respect to x as follows:

•The above derivative is referred to as the second derivative of y with respect to x. Higher derivatives
than the second-order are defined in a similar manner, for example:



•Example 9:

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14. Exercises
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15. Exercises
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16. Exercises
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