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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) 

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