Logarithmic Differentiation - Solved Examples (4)

Differentiate the following function by logarithmic differentiation

y =10^(cot(x))

• Take ln on both sides and simplify using logarithmic properties

ln(y) = ln(10^(cot(x)))

ln(y) = cot(x) * ln(10)

• Differentiate with respect to x

d/dx ln(y) = d/dx ( cot(x) * ln(10) )

y`/y = d/dx ( cot(x) * ln(10) ) 

•  Factor out ln(10) as it is a constant:

y`/y = ln(10) * d/dx cot(x) 

•  Differentiate cot(x)

y`/y = ln(10) * -csc^2(x)

• Cross multiply y on both sides and substitute the original value of y

y` = y * ln(10) * -csc^2(x)

y` = 10^(cot(x)) * ln(10) * -csc^2(x)