What is the derivative of csc^3(x)?

derivative of csc^3(x)

The derivative of

\csc^3(x)

-3 \cot(x)\csc^3(x)

.

Solution. To determine the derivative of

\csc^3(x)

, we will use the chain rule, i.e.:

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

where

F(x) = f(g(x)) = \csc^3(x)

f(u) = u^3

and

g(x) = \csc(x)

. The derivative of

f(u) = u^3

f'(u) = 3u^2

, and the derivative of

g(x) = \csc(x)

g'(x) = -\cot(x)\csc(x)

, which we have seen here. Therefore, we get:

\begin{align*} f'(g(x)) = f'(\csc(x)) = 3\csc^2(x) \quad \text{and} \quad g'(x) = -\cot(x)\csc(x). \end{align*}

Combining everything, we get:

\begin{align*} F'(x) &= f'(g(x))g'(x) \\ &= 3\csc^2(x) \cdot (-\cot(x)\csc(x)) \\ &= -3 \cot(x)\csc^3(x). \end{align*}

Therefore, the derivative of

\csc^3(x)

-3 \cot(x)\csc^3(x)