### 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

.

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

and differentiating, we get

.

Multiply both sides of this equation by y, getting

.

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

.

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

and differentiating, we get

(Get a common denominator and combine fractions on the right-hand side.)

(Factor out ex in the numerator.)

.

Multiply both sides of this equation by y, getting

(Combine the powers of x .)

.

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

.

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

and differentiating, we get

(Get a common denominator and combine fractions in the numerator.)

(Dividing by a fraction is the same as multiplying by its reciprocal.)

.

Multiply both sides of this equation by y, getting

(Combine the powers of (3x2+5) .)

.

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

.

Apply the natural logarithm to both sides of this equation getting

.

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

and differentiating, we get

(Get a common denominator and combine fractions on the right-hand side.)

.

Multiply both sides of this equation by y, getting

(Combine the powers of .)

.

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

.

Apply the natural logarithm to both sides of this equation and use the algebraic properties of logarithms, getting

.

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

and differentiating, we get

(Get a common denominator and combine fractions on the right-hand side.)

.

Multiply both sides of this equation by y, getting

(Divide out a factor of x .)

(Combine the powers of .)

.

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

.

Apply the natural logarithm to both sides of this equation and use the algebraic properties of logarithms, getting

.

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

and differentiating, we get

(Get a common denominator and combine fractions on the right-hand side.)

.

Multiply both sides of this equation by y, getting

(Combine the powers of .)

.

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

.

Apply the natural logarithm to both sides of this equation and use the algebraic properties of logarithms, getting

.

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

and differentiating, we get

(Divide out a factor of .)

(Get a common denominator and combine fractions on the right-hand side.)

.

Multiply both sides of this equation by y, getting

(Combine the powers of x .)