What is the Derivative of csc^2(x)?

Derivative of csc^2(x)

The derivative of

\csc^2(x)

-2\csc^2(x)\cot(x)

.

Solution. Let

F(x) = \csc^2(x)

f(u) = u^2

and

g(x) = \csc(x)

such that

\begin{align*} F(x) = f(g(x)). \end{align*}

Then we will use the chain rule to determine

F'(x)

\begin{align*} F'(x) = f'(g(x))g'(x). \end{align*}

We have already seen here that

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

and

f'(u) = \frac{d}{du} u^2 = 2u

. So we get:

\begin{align*} f'(g(x)) = 2g(x) = 2\csc(x) \quad \text{and} \quad g'(x) = -\csc(x)\cot(x). \end{align*}

Therefore, we get:

\begin{align*} F'(x) &= f'(g(x))g'(x) \\ &= 2\csc(x) \cdot (-\csc(x)\cot(x)) \\ &= -2\csc^2(x)\cot(x). \end{align*}

So, the derivative of

\csc^2(x)

-2\csc^2(x)\cot(x)