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
`(dy)/(dx)` is
`(y)/(1-y).`

Step-by-step explanation: We are given to find the value of
`(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

`(d)/(dx)\ln y=(d)/(dx)((\ln x)^2+2)\\\\\\\Rightarrow (1)/(y)(dy)/(dx)=(d)/(dx)(\ln x)^2+(d)/(dx)2\\\\\\\Rightarrow (1)/(y)(dy)/(dx)=2\ln x*(d)/(dx)\ln x+0\\\\\\\Rightarrow (1)/(y)(dy)/(dx)=2(\ln x)/(x)\\\\\\\Rightarrow (dy)/(dx)=(2y\ln x)/(x).`

Thus, the required value of
`(dy)/(dx)` in terms of x and y is
`(2y\ln x)/(x).`.