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प्रश्न
Find `(dy)/(dx)` if x + sin(x + y) = y – cos(x – y)
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उत्तर
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))`
- `d/(dx) (x)` = 1
- `d/(dx) (sin(x + y)) = cos(x + y) + cos(x + y)((dy)/(dx))`
- `- d/(dx) (y) = - (dy)/(dx)`
- `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))`
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