What is the derivative of 1/(csc^2(x)) ?

The derivative of 1/(csc^2(x)) is sin(2x)

What is the first derivative of 1/(csc^2(x)) ?

The first derivative of 1/(csc^2(x)) is sin(2x)

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

sin (2 x)

Solution steps

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

Simplify

\frac{1}{csc ^{2} ( x )} : sin^{2} (x)

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

Apply the chain rule:

2 sin (x) \frac{d}{d x} (sin (x))

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

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

= 2 sin (x) cos (x)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

= sin (2 x)

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