What is the derivative of - csc ( x ) ?

Question

What is the derivative of       -          csc              (        x        )            ?

Shiloh Hinton · Accepted Answer

Cosecant is the reciprocal of the sine function, so:

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

You should know that the derivative of

\frac{1}{x} = - \frac{1}{x^{2}}

Thus, by the chain rule:

\frac{1}{f (x)} = - \frac{1}{{f (x)}^{2} \cdot} \frac{d}{d x} f (x)

Applying this here:

- (- \frac{1}{\sin^{2} (x)} \cdot \frac{d}{d x} \sin (x))

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

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

This is already an answer, but it appears in most textbooks in a different form:

\frac{1}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)}

= \csc (x) \cdot \cot (x)

Answered question