# SOLUTIONS TO LOGARITHMIC DIFFERENTIATION

Jun 21st, 2015
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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

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