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

The derivative of (1-cos(x)/(1+sin(x))) is (1-cos(x)+sin(x))/(1+sin^2(x)+2sin(x))

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

The first derivative of (1-cos(x)/(1+sin(x))) is (1-cos(x)+sin(x))/(1+sin^2(x)+2sin(x))

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

What is the derivative of (1-cos(x)/(1+sin(x))) ?
The derivative of (1-cos(x)/(1+sin(x))) is (1-cos(x)+sin(x))/(1+sin^2(x)+2sin(x))
What is the first derivative of (1-cos(x)/(1+sin(x))) ?
The first derivative of (1-cos(x)/(1+sin(x))) is (1-cos(x)+sin(x))/(1+sin^2(x)+2sin(x))