Logarithmic Differentiation

If the power (or exponent) of a function of x is also, a function of x, or if the function is a root of a function is a root of a function, or a product of a number of functions, then it would be convenient for us to take logarithms with respect to the base ‘e’ and then differentiate both sides w.r.t.x.

Remark: (1) dy/dx (log y) = 1/y. dy/dx

(2) u^vx= e^{vlogu  [let z = uv, then log z = v log u ∴ z = evlogu , i.e. uv = evlogu]}

The following illustrations will make the idea clear.
Let Y = x

Then, taking logarithms of both the sides we get,
log y = x log x
Differentiating both the sides w.r.t.x, we get