Question

If ln y = (ln x)2 + 2, then find dy / dx in terms of x and y.

Answer 1

Answer:  The required value of $\dfrac{dy}{dx}$ is $\dfrac{y}{1-y}.$

Step-by-step explanation:   We are given to find the value of $\dfrac{dy}{dx}$ from the following equation :

$\ln y=(\ln x)^2+2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

Differentiating both sides of equation (i) with respect to x, we have

$\dfrac{d}{dx}\ln y=\dfrac{d}{dx}((\ln x)^2+2)\\\\\\\Rightarrow \dfrac{1}{y}\dfrac{dy}{dx}=\dfrac{d}{dx}(\ln x)^2+\dfrac{d}{dx}2\\\\\\\Rightarrow \dfrac{1}{y}\dfrac{dy}{dx}=2\ln x\times\dfrac{d}{dx}\ln x+0\\\\\\\Rightarrow \dfrac{1}{y}\dfrac{dy}{dx}=2\dfrac{\ln x}{x}\\\\\\\Rightarrow \dfrac{dy}{dx}=\dfrac{2y\ln x}{x}.$

Thus, the required value of $\dfrac{dy}{dx}$ in terms of x and y is $\dfrac{2y\ln x}{x}.$.