# Differentiation – Logarithmic

## Differentiation – Logarithmic

The function which can be evaluate by using logarithmic differentiation are of type y = [f (x)]g(x).

y = f₁ (x). f₂ (x). f₃ (x) …

$$y=\frac{{{f}_{1}}(x){{f}_{2}}(x){{f}_{3}}(x)}{{{g}_{1}}(x){{g}_{2}}(x){{g}_{3}}(x)}$$.

For this type of function take the logarithm and then differentiation.

Example 1: If xy. yx = 1, then the value of $$\frac{dy}{dx}$$ =?

Solution: Given that,

xy. yx = 1

Taking ‘ln’ on both sides

y lnx +x ln y = ln1

on differentiating both sides, we get

$$y\frac{d}{dx}(\ln \ x)+\left( \frac{d}{dx}.y \right).\ln \ x\ +\ \left( \frac{d}{dx}x \right).\ln \ y\ +x\left( \frac{d}{dx}.\ln y \right)=0$$.

$$y\frac{1}{x}+\ln x\ .\frac{dy}{dx}+1.\ln y+x.\frac{1}{y}.\frac{dy}{dx}=0$$.

$$\left[ \ln x+\frac{x}{y} \right].\frac{dy}{dx}=-\left[ \frac{y}{x}\ +\ln y \right]$$.

$$\frac{dy}{dx}=-\frac{(y+x\ln y)}{(x+y\ln x)}.\frac{y}{x}$$.

Example 2: If $$y={{x}^{{{(\log x)}^{\log \ \log x}}}}$$, then $$\frac{dy}{dx}$$ =?

Solution: Given that,

$$y={{x}^{{{(\log x)}^{\log \ \log x}}}}$$.

On taking log both sides

$$\log y=\log \left( {{x}^{{{(\log x)}^{\log \ \log x}}}} \right)$$.

$$\log y={{(\log x)}^{\log \ \log x}}.\log x$$.

Again, on taking log both sides

$$\log \ \log (y)=(\log \ \log x)\ \log \ \log x\ +\ \log \ \log x$$.

On differentiation with respect x

$$\frac{1}{\log y}.\frac{1}{y}\frac{dy}{dx}=2\log \log x.\frac{1}{\log x}.\frac{1}{x}+\frac{1}{\log x}.\frac{1}{x}$$,

$$\frac{dy}{dx}=\frac{y\log y}{x\log x}(2\log \ \log x\ +\ 1)$$.