Using Logarithmic Differentiation, find dy/dx given that

Question

y= −3 sec⁻¹(√ x) / 4^x(log₂ x) ^{sech x}

Answer 1

Solution:

y = −3 sec⁻¹(√ x) / [ 4^x(log₂ x) ^{sech x} ]

ln(y) = ln( −3 sec⁻¹(√ x) / [ 4^x(log₂ x) ^{sech x} ] )

All of the following rules were used to expand and "simplify":

ln(a*b) = ln(a) + ln(b) ln(a/b) = ln(a) - ln(b) ln(a^b) = b * ln(a)

ln(y) = ln(-1) + ln(3) + ln( sec⁻¹(√ x) ) - x ln(4) - sech x * [ ln( ln x ) - ln( ln 2 ) ]

e^iπ = - 1 ∴ ln( -1 ) = πi

ln(y) = πi + ln(3) + ln( sec⁻¹(√ x) ) - x ln(4) - sech x * [ ln( ln x ) - ln( ln 2 ) ]

y' / y = 0 + 0 + d/dx [ ln( sec⁻¹(√ x) ) ] - ln(4) - d/dx [ sech x * [ ln( ln x ) - ln( ln 2 ) ] ]