Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers) :
`(x^(2)cos(pi/4))/sinx`

Answer

To find the derivative of the function \( f(x) = \frac{x^2 \cos\left(\frac{\pi}{4}\right)}{\sin x} \), we will use the quotient rule. The quotient rule states that if you have a function \( \frac{u}{v} \), then the derivative is given by:

\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

### Step 1: Identify \( u \) and \( v \)
Here, we can identify:
- \( u = x^2 \cos\left(\frac{\pi}{4}\right) \)
- \( v = \sin x \)

### Step 2: Find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \)
First, we need to compute the derivatives of \( u \) and \( v \):
- Since \( \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \), we have:
\[ u = x^2 \cdot \frac{1}{\sqrt{2}} = \frac{x^2}{\sqrt{2}} \]
Thus,
\[ \frac{du}{dx} = \frac{2x}{\sqrt{2}} = \frac{2x}{\sqrt{2}} \]

- For \( v \):
\[ \frac{dv}{dx} = \cos x \]

### Step 3: Apply the Quotient Rule
Now, applying the quotient rule:
\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{\sin x \cdot \frac{2x}{\sqrt{2}} - \frac{x^2}{\sqrt{2}} \cdot \cos x}{\sin^2 x} \]

### Step 4: Simplify the Expression
Now we simplify the expression:
1. The numerator becomes:
\[ \sin x \cdot \frac{2x}{\sqrt{2}} - \frac{x^2}{\sqrt{2}} \cdot \cos x = \frac{2x \sin x - x^2 \cos x}{\sqrt{2}} \]

2. Therefore, the derivative is:
\[ \frac{d}{dx}\left(\frac{x^2 \cos\left(\frac{\pi}{4}\right)}{\sin x}\right) = \frac{2x \sin x - x^2 \cos x}{\sqrt{2} \sin^2 x} \]

### Final Answer
Thus, the derivative of the function is:
\[ \frac{2x \sin x - x^2 \cos x}{\sqrt{2} \sin^2 x} \]