Differentiate the following functions w.r.t. x:
`cosec""(2)/(3) x`

Answer

To differentiate the function \( y = \csc\left(\frac{2}{3} x\right) \) with respect to \( x \), we can follow these steps:

### Step 1: Identify the function
Let \( y = \csc\left(\frac{2}{3} x\right) \).

### Step 2: Use substitution
Let \( u = \frac{2}{3} x \). Then, we can rewrite \( y \) in terms of \( u \):
\[ y = \csc(u) \]

### Step 3: Differentiate with respect to \( u \)
The derivative of \( y \) with respect to \( u \) is:
\[ \frac{dy}{du} = -\csc(u) \cot(u) \]

### Step 4: Differentiate \( u \) with respect to \( x \)
Now, we differentiate \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{2}{3} \]

### Step 5: Apply the chain rule
Using the chain rule, we can find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = -\csc(u) \cot(u) \cdot \frac{2}{3} \]

### Step 6: Substitute back \( u \)
Now, substitute \( u = \frac{2}{3} x \) back into the equation:
\[ \frac{dy}{dx} = -\csc\left(\frac{2}{3} x\right) \cot\left(\frac{2}{3} x\right) \cdot \frac{2}{3} \]

### Final Answer
Thus, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{2}{3} \csc\left(\frac{2}{3} x\right) \cot\left(\frac{2}{3} x\right) \]

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