Question

Find dy/dx given that y = sin x / 1 + cos x​

Answer 1

`(1)/(1 + \cos(x) )`

Explanation:

`y = ( \sin(x) )/(1 + \cos(x) )`

differentiating numerator wrt x :-

(sinx)' = cos x

differentiating denominator wrt x :-

(1 + cos x)' = (cosx)' = - sinx

• Let's say the denominator was "v" and the numerator was "u"

`((u)/(v) )' = \frac{v. \: (u)' - u.(v)' }{ {v}^(2) }`

here,

• since u is the numerator u= sinx and u = cos x
• v(denominator) = 1 + cos x; v' = - sinx

`= \frac{((1 + \cos \: x) \cos \: x )- (\sin \: x. ( - \sin \: x) ) }{( {1 + \cos(x)) }^(2) }`

`= \frac{ \cos(x) + \cos {}^(2) (x) + \sin {}^(2) (x) }{(1 + \cos \: x) {}^(2) }`

since cos²x + sin²x = 1

`= \frac{ \cos \: x + 1}{(1 + \cos \: x) {}^(2) }`

diving numerator and denominator by 1 + cos x

`= (1)/(1 + \cos(x) )`