Find the derivatives w.r.t. x:

• 2 cosec^-1(3x) + 3 cosec²x

To find the derivative of the function \( f(x) = 2 \csc^{-1}(3x) + 3 \csc(2x) \) with respect to \( x \), we will differentiate each term separately.

Step 1: Differentiate \( 2 \csc^{-1}(3x) \)

The derivative of \( \csc^{-1}(u) \) is given by:

• \( \frac{d}{dx} \csc^{-1}(u) = \frac{u'}{|u| \sqrt{u^2 - 1}} \)

Here, \( u = 3x \), so \( u' = 3 \). Thus, the derivative becomes:

\( \frac{d}{dx} \left( 2 \csc^{-1}(3x) \right) = 2 \cdot \frac{3}{|3x| \sqrt{(3x)^2 - 1}} = \frac{6}{|3x| \sqrt{9x^2 - 1}} \)

Step 2: Differentiate \( 3 \csc(2x) \)

The derivative of \( \csc(u) \) is:

• \( \frac{d}{dx} \csc(u) = -\csc(u) \cot(u) \cdot u' \)

For \( u = 2x \), we have \( u' = 2 \). Therefore, the derivative is:

\( \frac{d}{dx} \left( 3 \csc(2x) \right) = 3 \cdot \left( -\csc(2x) \cot(2x) \cdot 2 \right) = -6 \csc(2x) \cot(2x) \)

Final Derivative

Combining both parts, the derivative of the function is:

\( f'(x) = \frac{6}{|3x| \sqrt{9x^2 - 1}} - 6 \csc(2x) \cot(2x)

This expression gives the rate of change of the original function with respect to \( x \).