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Lecture

Logarithmic Differentiation Solution.pdf

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Department
Mathematics
Course
MAT157Y1
Professor
Dan Dolderman
Semester
Fall

Description
11/ 8/ 12 No Tit le 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 = x . Apply the naturallogarithmto both sides ofthis equation getting . Differentiate both sides ofthis 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 ofthis equation by y, getting . Click HERE to return to the list ofproblems. 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 (e ) . Apply the naturallogarithmto both sides ofthis equation getting www. m at h. ucdavis. edu/ ~kouba/ CalcO neDI RECTO RY/ logdif f soldir ect or y/ LogDif f Sol. ht m l#SO LUTI O N 4 1/ 9 11/ 8/ 12 No Tit le . Differentiate both sides ofthis 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 e in the numerator.) . Multiply both sides ofthis equation by y, getting (Combine the powers ofx .) . Click HERE to return to the list ofproblems. www. m at h. ucdavis. edu/ ~kouba/ CalcO neDI RECTO RY/ logdif f soldir ect or y/ LogDif f Sol. ht m l#SO LUTI O N 4 2/ 9 11/ 8/ 12 No Tit le 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 = (3x +5)2 1/x . Apply the naturallogarithmto both sides ofthis equation getting . Differentiate both sides ofthis 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.) www. m at h. ucdavis. edu/ ~kouba/ CalcO neDI RECTO RY/ logdif f soldir ect or y/ LogDif f Sol. ht m l#SO LUTI O N 4 3/ 9 11/ 8/ 12 No Tit le . Multiply both sides ofthis equation by y, getting 2 (Combine the powers of(3x +5) .) . Click HERE to return to the list ofproblems. 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 naturallogarithmto both sides ofthis equation getting . Differentiate both sides ofthis 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.) www. m at h. ucdavis. edu/ ~kouba/ CalcO neDI RECTO RY/ logdif f soldir ect or y/ LogDif f Sol. ht m l#SO LUTI O N 4 4/ 9 11/ 8/ 12 No Tit le . Multiply both sides ofthis equation by y, getting
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