Calculate `(dy)/(dx)` for the following :-

Options

A. `y=cosx^(3)`
B. `y=sin((x)/(2))`
C. `y=log_(e)2x`
D. `y=e^(-x)`

Answer

To calculate \(\frac{dy}{dx}\) for the given functions, we will apply the rules of differentiation step by step for each function.

### Step 1: Differentiate \(y = \cos(x^3)\)

1. **Identify the function**: \(y = \cos(x^3)\).
2. **Apply the chain rule**: The derivative of \(\cos(u)\) is \(-\sin(u)\), where \(u = x^3\).
3. **Differentiate the inner function**: The derivative of \(x^3\) is \(3x^2\).
4. **Combine results**:
\[ \frac{dy}{dx} = -\sin(x^3) \cdot \frac{d}{dx}(x^3) = -\sin(x^3) \cdot 3x^2 = -3x^2 \sin(x^3). \]

### Step 2: Differentiate \(y = \sin\left(\frac{x}{2}\right)\)

1. **Identify the function**: \(y = \sin\left(\frac{x}{2}\right)\).
2. **Apply the chain rule**: The derivative of \(\sin(u)\) is \(\cos(u)\), where \(u = \frac{x}{2}\).
3. **Differentiate the inner function**: The derivative of \(\frac{x}{2}\) is \(\frac{1}{2}\).
4. **Combine results**:
\[ \frac{dy}{dx} = \cos\left(\frac{x}{2}\right) \cdot \frac{d}{dx}\left(\frac{x}{2}\right) = \cos\left(\frac{x}{2}\right) \cdot \frac{1}{2} = \frac{1}{2} \cos\left(\frac{x}{2}\right). \]

### Step 3: Differentiate \(y = \ln(2x)\)

1. **Identify the function**: \(y = \ln(2x)\).
2. **Apply the chain rule**: The derivative of \(\ln(u)\) is \(\frac{1}{u}\), where \(u = 2x\).
3. **Differentiate the inner function**: The derivative of \(2x\) is \(2\).
4. **Combine results**:
\[ \frac{dy}{dx} = \frac{1}{2x} \cdot \frac{d}{dx}(2x) = \frac{1}{2x} \cdot 2 = \frac{1}{x}. \]

### Step 4: Differentiate \(y = e^{-x}\)

1. **Identify the function**: \(y = e^{-x}\).
2. **Apply the chain rule**: The derivative of \(e^u\) is \(e^u\), where \(u = -x\).
3. **Differentiate the inner function**: The derivative of \(-x\) is \(-1\).
4. **Combine results**:
\[ \frac{dy}{dx} = e^{-x} \cdot \frac{d}{dx}(-x) = e^{-x} \cdot (-1) = -e^{-x}. \]

### Final Results

1. \(\frac{dy}{dx} = -3x^2 \sin(x^3)\)
2. \(\frac{dy}{dx} = \frac{1}{2} \cos\left(\frac{x}{2}\right)\)
3. \(\frac{dy}{dx} = \frac{1}{x}\)
4. \(\frac{dy}{dx} = -e^{-x}\)