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

The derivative of (csc(x)/(1+csc(x))) is -(cot(x)csc(x))/((1+csc(x))^2)

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

The first derivative of (csc(x)/(1+csc(x))) is -(cot(x)csc(x))/((1+csc(x))^2)

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\frac{d}{d x} (\frac{csc ( x )}{1 + csc ( x )})

- \frac{cot ( x ) csc ( x )}{( 1 + csc ( x ) ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{csc ( x )}{1 + csc ( x )})

Apply the Quotient Rule:

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

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

\frac{d}{d x} (csc (x)) = - cot (x) csc (x)

\frac{d}{d x} (1 + csc (x)) = - cot (x) csc (x)

= \frac{( - cot ( x ) csc ( x ) ) ( 1 + csc ( x ) ) - ( - cot ( x ) csc ( x ) ) csc ( x )}{( 1 + csc ( x ) ) ^{2}}

Simplify

\frac{( - cot ( x ) csc ( x ) ) ( 1 + csc ( x ) ) - ( - cot ( x ) csc ( x ) ) csc ( x )}{( 1 + csc ( x ) ) ^{2}} : - \frac{cot ( x ) csc ( x )}{( 1 + csc ( x ) ) ^{2}}

= - \frac{cot ( x ) csc ( x )}{( 1 + csc ( x ) ) ^{2}}

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