Derivative of csc x

How to find the derivative of the given function by using the derivative of csc x: formula, 1 example, and its solution.

Formula

Formula

The derivative of csc x, [csc x]',
is -csc x cot x.

Example

Example

Solution

y = csc (1 - x2) is a composite function of
y = csc (whole) and (whole) = 1 - x2.

So the derivative of the composite function is,
the derivative of the outer function csc (1 - x2), -csc (1 - x2) cot (1 - x2)
times,
the derivative of the inner function 1 - x2, -2x1.

-(-2x1) = 2x

So the right side is 2x csc (1 - x2) cot (1 - x2).

So 2x csc (1 - x2) cot (1 - x2) is the derivative of csc (1 - x2).