f(x) = (sinx)˟
ln[f(x)] = ln(sinx)˟
ln[f(x)] = xln(sinx)
f'(x) / f(x) = ln(sinx) + xcosx / sinx
f'(x) = f(x)[ln(sinx) + xcosx / sinx]
f'(x) = (sinx)˟[ln(sinx) + xcosx / sinx]
f'(x) = (sinx)˟[ln(sinx) + xcotx]

.Q. Differentiate y = (sinx)^x
Ans : Given : y = (sinx)^x
Apply "log" on both sides .
∴ log(y) = log((sinx)^x)
Apply : log(a^n) = n log(a)
⇒ log(y) = x log(sinx)
Differentiate both sides wrt 'x'

⇒ d{log(y)} / dx = d{x*log(sinx)} / dx
Apply Product rule of Differentiation in RHS :
For any of the functions f and g,
(fg)' = f' g + f g',

⇒[1/y] * (dy/dx) = { d(x)/dx * log(sinx) + x * d(log(sinx))/dx }
⇒[1/y] * (dy/dx) = {1*log(sinx) + x * [ 1/sinx * [ d(sinx)/dx] ] }
⇒[1/y] * (dy/dx) = {log(sinx) + x*[cosx / sinx] }
⇒[1/y] * (dy/dx) = {log(sinx) + x*cotx}
⇒ (dy/dx) = {log(sinx) + x*cotx} * y
∴ (dy/dx) = {log(sinx) + x*cotx} * (sinx)^x

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