Use logarithmic differentiation to find the derivative of the function. y = (sin(5x))ln(x)

Solution:

Logarithmic differentiation is based on the logarithm properties and the chain rule of differentiation.

Property of logarithmic functions to be used in the given problem

Some of the properties of the logarithmic terms are

(i) log_a^b  = x ⇒ a^x = b

(ii) log_m^a = log_m^b = log_m^ab

(iii) log(a/b) = loga - logb

(iv) log a^m = m loga

(v) log_a^b = 1/log_b^a

Let y = (sin (5x)) ln(x)

Taking a log on both sides,

ln y = ln [(sin (5x)) ln(x)] 

ln y = ln (sin (5x)) + ln (ln (x))

Differentiating with respect to x

(1/y) (dy/dx) = (1/ sin 5x)(cos 5x)(5) + (1/ lnx )(1/x)

dy/dx = y{5cot (5x) + (1/ x ln x)}

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Use logarithmic differentiation to find the derivative of the function. y = (sin(5x))ln(x)

Summary:

Using logarithms the derivative of y = (sin(5x))ln(x) is = y{5cot (5x) + (1/ x lnx)}