Question

Find dy/dx if y=cot(x+y)​

Answer 1

Given:

The equation is

`y=\cot(x+y)`

To find:

The value of
`(dy)/(dx)`.

Solution:

We have,

`y=\cot(x+y)`

Differentiate with respect to x.

`(dy)/(dx)=(d)/(dx)\cot(x+y)`

`(dy)/(dx)=-\text{cosec}^2(x+y)(d)/(dx)(x+y)`

`(dy)/(dx)=-\text{cosec}^2(x+y)(1+(dy)/(dx))`

`(dy)/(dx)=-\text{cosec}^2(x+y)-\text{cosec}^2(x+y)(dy)/(dx)`

`(dy)/(dx)+\text{cosec}^2(x+y)(dy)/(dx)=-\text{cosec}^2(x+y)`

`(dy)/(dx)(1+\text{cosec}^2(x+y))=-\text{cosec}^2(x+y)`

`(dy)/(dx)=-\frac{\text{cosec}^2(x+y)}{1+\text{cosec}^2(x+y)}`

Therefore,
`(dy)/(dx)=-\frac{\text{cosec}^2(x+y)}{1+\text{cosec}^2(x+y)}`.