Differentiate:`(2^(x)cotx)/(sqrtx)`

Answer

To differentiate the function \( y = \frac{2^x \cot x}{\sqrt{x}} \), we will use the quotient rule. The quotient rule states that if you have a function in the form \( \frac{f(x)}{g(x)} \), the derivative is given by:

\[ \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{g(x) f'(x) - f(x) g'(x)}{(g(x))^2} \]

### Step 1: Identify \( f(x) \) and \( g(x) \)

Let:
- \( f(x) = 2^x \cot x \)
- \( g(x) = \sqrt{x} \)

### Step 2: Differentiate \( f(x) \) and \( g(x) \)

Now we need to find \( f'(x) \) and \( g'(x) \).

#### Differentiate \( g(x) \):

\[ g(x) = \sqrt{x} = x^{1/2} \]
Using the power rule:
\[ g'(x) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}} \]

#### Differentiate \( f(x) \):

To differentiate \( f(x) = 2^x \cot x \), we will use the product rule:
\[ f'(x) = \frac{d}{dx}(2^x) \cot x + 2^x \frac{d}{dx}(\cot x) \]

1. Differentiate \( 2^x \):
\[ \frac{d}{dx}(2^x) = 2^x \ln(2) \]

2. Differentiate \( \cot x \):
\[ \frac{d}{dx}(\cot x) = -\csc^2 x \]

Putting it all together:
\[ f'(x) = 2^x \ln(2) \cot x + 2^x (-\csc^2 x) = 2^x \left( \ln(2) \cot x - \csc^2 x \right) \]

### Step 3: Apply the Quotient Rule

Now we can apply the quotient rule:
\[ y' = \frac{g(x) f'(x) - f(x) g'(x)}{(g(x))^2} \]

Substituting \( f(x) \), \( f'(x) \), \( g(x) \), and \( g'(x) \):
\[ y' = \frac{\sqrt{x} \left( 2^x \left( \ln(2) \cot x - \csc^2 x \right) \right) - (2^x \cot x) \left( \frac{1}{2\sqrt{x}} \right)}{(\sqrt{x})^2} \]

### Step 4: Simplify the Expression

The denominator simplifies to:
\[ (\sqrt{x})^2 = x \]

Thus:
\[ y' = \frac{\sqrt{x} \left( 2^x \left( \ln(2) \cot x - \csc^2 x \right) \right) - \frac{2^x \cot x}{2\sqrt{x}}}{x} \]

### Step 5: Combine Terms

Now, we can combine the terms in the numerator:
\[ y' = \frac{2^x \sqrt{x} \left( \ln(2) \cot x - \csc^2 x \right) - \frac{2^x \cot x}{2\sqrt{x}}}{x} \]

This is the derivative of the function.

### Final Answer

\[ y' = \frac{2^x \left( \sqrt{x} \left( \ln(2) \cot x - \csc^2 x \right) - \frac{\cot x}{2\sqrt{x}} \right)}{x} \]