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 .) 