LOGARITHMIC DIFFERENTIATION
LOGARITHMIC DIFFERENTIATION
The following problems illustrate the process of logarithmic differentiation. It is a means of differentiating algebraically complicated functions or functions for which the ordinary rules of differentiation do not apply. For example, in the problems that follow, you will be asked to differentiate expressions where a variable is raised to a variable power. An example and two COMMON INCORRECT SOLUTIONS are :
1.) [pic]
and
2.) [pic].
BOTH OF THESE SOLUTIONS ARE WRONG because the ordinary rules of differentiation do not apply. Logarithmic differentiation will provide a way to differentiate a function of this type. It requires deft algebra skills and careful use of the following unpopular, but well-known, properties of logarithms. Though the following properties and methods are true for a logarithm of any base, only the natural logarithm (base e, where e [pic]), [pic], will be used in this problem set.
PROPERTIES OF THE NATURAL LOGARITHM
1. [pic].
2. [pic].
3. [pic].
4. [pic].
5. [pic].
6. [pic].
AVOID THE FOLLOWING LIST OF COMMON MISTAKES
1. [pic].
2. [pic].
3. [pic].
4. [pic].
5. [pic].
The following problems range in difficulty from average to challenging
SOLUTIONS TO LOGARITHMIC DIFFERENTIATION
SOLUTION 1 : Because a variable is raised to a variable power in this function, the ordinary rules of differentiation DO NOT APPLY ! The function must first be revised before a derivative can be taken. Begin with
y = xx .
Apply the natural logarithm to both sides of this equation getting
[pic]
[pic].
Differentiate both sides of this equation. The left-hand side requires the chain rule since y represents a function of x . Use the product rule on the right-hand side. Thus, beginning with
[pic]
and differentiating, we get
[pic]
[pic].
Multiply both sides of this equation by y, getting
[pic].
SOLUTION 2 : Because a variable is raised to a variable power in this function, the ordinary rules of differentiation DO NOT APPLY ! The function must first be revised before a derivative can be taken. Begin with
y = x(ex) .
Apply the natural logarithm to both sides of this equation getting
[pic]
[pic].
Differentiate both sides of this equation. The left-hand side requires the chain rule since y represents a function of x . Use the product rule on the right-hand side. Thus, beginning with
[pic]
and differentiating, we get
[pic]
(Get a common denominator and combine fractions on the right-hand side.)
[pic]
[pic]
[pic]
(Factor out ex in the numerator.)
[pic].
Multiply both sides of this equation by y, getting
[pic]
[pic]
(Combine the powers of x .)
[pic].
SOLUTION 3 : Because a variable is raised to a variable power in this function, the ordinary rules of differentiation DO NOT APPLY ! The function must first be revised before a derivative can be taken. Begin with
y = (3x2+5)1/x .
Apply the natural logarithm to both sides of this equation getting
[pic]
[pic]
[pic].
Differentiate both sides of this equation. The left-hand side requires the chain rule since y represents a function of x . Use the quotient rule and the chain rule on the right-hand side. Thus, beginning with
[pic]
and differentiating, we get
[pic]
(Get a common denominator and combine fractions in the numerator.)
[pic]
(Dividing by a fraction is the same as multiplying by its reciprocal.)
[pic]
[pic].
Multiply both sides of this equation by y, getting
[pic]
[pic]
(Combine the powers of (3x2+5) .)
[pic].
SOLUTION 4 : Because a variable is raised to a variable power in this function, the ordinary rules of differentiation DO NOT APPLY ! The function must first be revised before a derivative can be taken. Begin with
[pic].
Apply the natural logarithm to both sides of this equation getting
[pic]
[pic].
Differentiate both sides of this equation. The left-hand side requires the chain rule since y represents a function of x . Use the product rule and the chain rule on the right-hand side. Thus, beginning with truein [pic]
and differentiating, we get
[pic]
(Get a common denominator and combine fractions on the right-hand side.)
[pic]
[pic].
Multiply both sides of this equation by y, getting
[pic]
[pic]
(Combine the powers of [pic].)
[pic].
SOLUTION 5 : Because a variable is raised to a variable power in this function, the ordinary rules of differentiation DO NOT APPLY ! The function must first be revised before a derivative can be taken. Begin with
[pic].
Apply the natural logarithm to both sides of this equation and use the algebraic properties of logarithms, getting
[pic]
[pic]
[pic].
Differentiate both sides of this equation. The left-hand side requires the chain rule since y represents a function of x . Use the product rule and the chain rule on the right-hand side. Thus, beginning with
[pic]
and differentiating, we get
[pic]
[pic]
(Get a common denominator and combine fractions on the right-hand side.)
[pic]
[pic].
Multiply both sides of this equation by y, getting
[pic]
[pic]
(Divide out a factor of x .)
[pic]
(Combine the powers of [pic].)
[pic].
SOLUTION 6 : Because a variable is raised to a variable power in this function, the ordinary rules of differentiation DO NOT APPLY ! The function must first be revised before a derivative can be taken. Begin with
[pic].
Apply the natural logarithm to both sides of this equation and use the algebraic properties of logarithms, getting
[pic]
[pic]
[pic]
[pic]
[pic].
Differentiate both sides of this equation. The left-hand side requires the chain rule since y represents a function of x . Use the product rule and the chain rule on the right-hand side. Thus, beginning with
[pic]
and differentiating, we get
[pic]
[pic]
(Get a common denominator and combine fractions on the right-hand side.)
[pic]
[pic]
[pic].
Multiply both sides of this equation by y, getting
[pic]
[pic]
(Combine the powers of [pic].)
[pic].
SOLUTION 7 : Because a variable is raised to a variable power in this function, the ordinary rules of differentiation DO NOT APPLY ! The function must first be revised before a derivative can be taken. Begin with
[pic].
Apply the natural logarithm to both sides of this equation and use the algebraic properties of logarithms, getting
[pic]
[pic]
[pic]
[pic].
Differentiate both sides of this equation. The left-hand side requires the chain rule since y represents a function of x . Use the product rule and the chain rule on the right-hand side. Thus, beginning with
[pic]
and differentiating, we get
[pic]
(Divide out a factor of [pic].)
[pic]
(Get a common denominator and combine fractions on the right-hand side.)
[pic]
[pic].
Multiply both sides of this equation by y, getting
[pic]
[pic]
(Combine the powers of x .)
[pic]
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