EXAMPLES OF LOGARITHMIC DIFFERENTIATION

Differentiate each of the following with respect to x.

Example 1 :

y = [sin x cos (x²)]/[x³ + lnx]

Solution :

Take  logarithm on both sides

lny = ln[sinxcos(x²)]/[x³ + lnx]

lny = ln[sinxcos(x²)] - ln(x³ + lnx)

lny = ln(sinx) + lncos(x²) - ln(x³ + lnx) ----(1)

Derivative of ln y :

= (1/y)y'

= y'/y

Derivative of ln(sin x) :

= (1/sinx)cosx

= cosx/sinx

= cot x

Derivative of lncos (x²) :

= (1/cosx²) - sinx²(2x)

= -2xtan x²

Derivative of ln(x³ + ln x) :

= [1/(x³ + lnx)][3x² + 1/x]

= [1/(x³ + lnx)][(3x³ + 1)/x]

= [(3x³ + 1)/x(x³ + lnx)]

(1) :

y'/y = cotx - 2xtanx² - [(3x³ + 1)/x(x³ + lnx)]

ny' = y[cotx - 2xtanx² - [(3x³ + 1)/x(x³ + lnx)]]

ny' = ([sinxcos(x²)]/[x³ + lnx])[cotx - 2xtanx²- [(3x³ + 1)/x(x³ + lnx)]]

Example 2 :

y = (x² + 2)(x + √2)/√(x + 4)

Solution :

lny = ln{[(x² + 2)(x + √2)]/√(x + 4)

lny = ln(x² + 2) + ln(x + √2) - ln √(x + 4) ----(1)

Derivative of lny :

= (1/y)y'

= y'/y

Derivative of ln(x² + 2) :

= (1/(x² + 2))(2x)

= 2x/(x²+2)

Derivative of ln(x + √2) :

= 1/(x + √2)

Derivative of ln√(x + 4) :

= (1/√(x + 4)) ⋅ 1/2√(x + 4)

=  1/2(x + 4)

(1) :

y'/y = [2x/(x² + 2)] + 1/(x + √2) + [1/2(x + 4)]

y' =  y[2x/(x² + 2)] + 1/(x + √2) + [1/2(x + 4)]

y' = ((x² + 2)(x + √2)/√(x + 4))[2x/(x² + 2)] + 1/(x + √2) +[1/2(x + 4)]

Example 3 :

y = x^√x

Solution :

y = x^√x

Take logarithm on both sides.

ln y = ln (x^√x)

ln y = √x ln (x)

Differentiate with respect to x.

u = √x and u' = 1/2√x 

v = lnx and v' = 1/x

(1/y) y' = √x (1/x) + ln (x)(1/2√x)

= (√x/x) + ln x/2√x

= (1/√x) + ln x/2√x

(y'/y) = (2 + ln x)/2√x

y' = x^√x(2 + ln x)/2√x

Example 4 :

y = x^sinx

Solution :

y = x^sinx

Take logarithm on both sides.

ln y = ln x^sinx

ln y = sin x ln x

Differentiate with respect to x.

u = sin x and u' = cos x

v = ln x and v' = 1/x

(1/y)y' = sin x (1/x) + ln x(cos x)

y'/y  =  sin x/x + cos x (ln x)

y' = y[(sin x/x) + cos x(ln x)]

y' = x^sinx[(sin x/x) + cos x(ln x)]

Example 5 :

y = (ln x)^cosx

Solution :

y = (ln x)^cosx

Take logarithm on both sides.

ln y = ln ((ln x)^cosx)

ln y = cosx ln (ln x)

Differentiate with respect to x.

(1/y)y' = cos x (1/ln x)(1/x) + ln(ln x)(-sinx)

y' = y[cos x/x ln x) - sin x(ln (ln x)]

y' = (lnx)^cosx(cos x/x ln x) - sin x(ln(ln x))

Example 6 :

y = x^{log x} + (log x)^x

Solution :

y = x^{log x} + (log x)^x

Part 1 :

Let y = x^{log x}

log y = log (x^{log x})

log y = log x (log x)

log y = (log x)²

log y = 2(log x)

Differentiating with respect to x, we get

(1/y) dy/dx = 2(1/x)

dy/dx = y(2/x)

Applying the value of y, we get

= x^{log x}(2/x)

Part 2 :

y = (log x)^x

Taking log on both sides

log y = log ((log x)^x)

log y = x log (log x)

(1/y)(dy/dx) = x (1/log x)(1/x) + log (log x) (1)

dy/dx = y[1/log x + log (log x)]

Applying the value of y, we get

= (log x)^x [1/log x + log (log x)]

Example 7 :

y = tan^-1 (6x/(1 - 9x²))

Solution :

y = tan^-1 (6x/(1 - 9x²))

y = tan^-1 (2(3x)/(1 - (3x)²))

Let 3x = tan θ

tan 2 θ = 2 tan θ / (1 - tan²θ)

y = tan^-1 (2 tan θ/(1 - tan²θ))

y = tan^-1 (tan 2 θ)

y = 2 θ

deriving value of θ from 3x = tan θ

θ = tan^-1(3x)

y = 2 tan^-1(3x)

dy/dx = 2 (1/(1 + (3x)²) (3)

dy/dx = 6/(1 + (3x)²

Example 8 :

y = tan^-1 (√(1 - cos x)/(1 + cos x))

Solution :

y = tan^-1 (√(1 - cos x)/(1 + cos x))

From cos 2 x = 1 - 2 sin² x

2sin² x = 1 - cos 2x

Dividing θ by 2, we get

2sin² (x/2) = 1 - cos x ----(1)

From cos 2 x = 2 cos² x - 1

2 cos² x/2 = 1 + cos x ----(2)

(1) / (2)

(1 - cos x)/(1 + cos x) = 2sin² (x/2) / 2 cos² x/2

(1 - cos x)/(1 + cos x) = tan² (x/2)

Taking square roots on both sides

√(1 - cos x)/(1 + cos x) = √tan² (x/2)

√(1 - cos x)/(1 + cos x) = tan (x/2)

Applying the above value, we get

y = tan^-1 (√(1 - cos x)/(1 + cos x))

y = tan^-1 (tan (x/2))

y = x/2

dy/dx = 1/2

Example 9 :

y = sin^-1 (3x - 4x³)

Solution :

y = sin^-1 (3x - 4x³)

Let x = sin θ

y = sin^-1 (3 sin θ - 4(sin³θ))

y = sin^-1 (sin 3θ)

y = 3θ

Deriving value of θ

θ = sin^-1(x)

y = 3sin^-1(x)

Differentiating with respect to x, we get

dy/dx = 3(1 / √(1 - x²))

dy/dx = 3/√(1 - x²)

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