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amid [387]
3 years ago
14

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

Mathematics
1 answer:
kobusy [5.1K]3 years ago
6 0

Answer:

\frac{1}{1 +  \cos(x) }

Step-by-step explanation:

y =  \frac{ \sin(x) }{1 +  \cos(x) }

<u>differentiating numerator wrt x :-</u>

(sinx)' = cos x

<u>differentiating denominator wrt x :- </u>

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

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

(\frac{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

=  \frac{1}{1 +  \cos(x) }

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