What is the derivative of (cos(x)/(sin(x)+1)) ?

The derivative of (cos(x)/(sin(x)+1)) is -1/(1+sin(x))

What is the first derivative of (cos(x)/(sin(x)+1)) ?

The first derivative of (cos(x)/(sin(x)+1)) is -1/(1+sin(x))

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

- \frac{1}{1 + sin ( x )}

Solution steps

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( cos ( x ) ) ( sin ( x ) + 1 ) - \frac{d}{d x} ( sin ( x ) + 1 ) cos ( x )}{( sin ( x ) + 1 ) ^{2}}

\frac{d}{d x} (cos (x)) = - sin (x)

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

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

Simplify

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

= - \frac{1}{1 + sin ( x )}

What is the derivative of (cos(x)/(sin(x)+1)) ?
The derivative of (cos(x)/(sin(x)+1)) is -1/(1+sin(x))
What is the first derivative of (cos(x)/(sin(x)+1)) ?
The first derivative of (cos(x)/(sin(x)+1)) is -1/(1+sin(x))