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MAT157Y1
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Dan Dolderman
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Mathematics

MAT157Y1

Dan Dolderman

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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