LOGARITHMIC DIFFERENTIATION PROBLEMS AND SOLUTIONS

Logarithmic differentiation is a method of finding derivatives of some complicated functions, using the properties of logarithms.

There are cases in which differentiating the logarithm of a given function is easier than differentiating the function as it is.

Step 1 : 

Take logarithm on both sides of the given equation. 

Step 2 : 

Use the properties of logarithm. 

Step 3 : 

Differentiate with respect to x and solve for dy/dx. 

Differentiate each of the following with respect to x.

Problem 1 :

xy = yx 

Solution :

xy = yx 

Taking logarithm on both sides.

logxy = logyx

ylogx = xlogy

Differentiate with respect to x. 

y(1/x) + logx(dy/dx) = x(1/y)(dy/dx) + logy(1)

y/x + logx(dy/dx) = (x/y)(dy/dx) + logy

logx(dy/dx) - (x/y)(dy/dx) = logy - y/x

(logx - x/y)(dy/dx) = (xlogy - y)/x

[(ylogx - x)/y](dy/dx) = (xlogy - y)/x

dy/dx = (y/x)[(xlogy - y)/(ylogx - x)]

Problem 2 :

y = (cosx)logx

Solution :

y = (cosx)logx

logy = log[(cosx)logx]

logy = (logx)log(cosx)

Differentiate with respect to x. 

(1/y)(dy/dx) = logx(1/cosx)(-sinx) + log(cosx)(1/x)

(1/y)(dy/dx) = -logxtanx + log(cosx)/x

(dy/dx) = y[-logxtanx + log(cosx)/x]

dy/dx = (cosx)logx[-logxtanx + log(cos x)/x]

Problem 3 :

(x2/a2) + (y2/b2) = 1

Solution :

Its an implicit function. Since the function is not a complicated one, we don't have to use logarithm to find derivative.

(x2/a2) + (y2/b2) = 1

Differentiate with respect to x. 

(2x/a2) + (2y/b2)(dy/dx) = 0

(2y/b2)(dy/dx) = -(2x/a2)

(dy/dx) = -(b2/a2)(x/y)

(dy/dx) = -(b2x/a2y)

Problem 4 :

√(x2 + y2) = tan-1(y/x)

Solution :

√(x2 + y2) = tan-1(y/x)

Differentiate √(x2 + y2) with respect to x.

= [1/2√(x2+ y2)][2x + 2y(dy/dx)]

= [x + y(dy/dx)]/√(x2+y2) ----(1)

Differentiate tan-1(y/x) with respect to x.

  = 1/[1 + (y/x)2](-y/x2) + (1/x)(dy/dx)

= [x2/(x2 + y2)][x(dy/dx) - y]/x2

(x(dy/dx) - y)/(x2 + y2----(2)

(1) = (2)

[x +  y(dy/dx)]/√(x2+y2) = (x(dy/dx) - y)/(x2 + y2)

√(x+ y2)[x + y(dy/dx)] = x(dy/dx) - y

x√(x+ y2) + y√(x+ y2)(dy/dx) - x(dy/dx)  =  -y

(dy/dx)(y√(x+ y2) - x) = -y - x√(x+ y2)

(dy/dx) = (x√(x2+y2) + y)/(x - y√(x2+y2))

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