Question

Find `(dy)/(dx)` if x + sin(x + y) = y – cos(x – y)

Answer 1

Given: x + sin(x + y) = y – cos(x – y)

To Find: Derivative of x + sin(x + y) = y – cos(x – y)

Step-by-step explanation:

Apply the sum/Difference Rule: (f ± g)' = f' ± g'

= `d/(dx) (x) + d/(dx) (sin(x + y)) - d/(dx) (y) + d/(dx) (cos(x - y))`

1. `d/(dx) (x)` = 1
2. `d/(dx) (sin(x + y)) = cos(x + y) + cos(x + y)((dy)/(dx))`
3. `- d/(dx) (y) = - (dy)/(dx)`
4. `d/(dx) (cos(x - y))`

Adding up all, we get;

⇒ 0 = `1 + cos(x + y)(1 + d/(dx) (y)) - d/(dx) (y) - sin(x - y)(1 - d/(dx) (y))`

Taking `(dy)/(dx)` on the left-hand side of the equation, we get:

`(dy)/(dx) = (1 + cos(x + y) - sin(x - y))/(- cos(x + y) + 1- sin(x - y))`

Hence, the derivative of the given equation is: `(1 + cos(x + y) - sin(x - y))/(1 - cos(x + y) - sin(x - y))`