Differentiate the following w.r.t.x

Options

A. `y=x^(2)+5`
B. `y=2e^(3)`
C. `y=(x+5)^(-1//2)`
D. `y=5x^(3//2)`

Answer

To differentiate the given functions with respect to \( x \), we will follow the rules of differentiation step by step for each function.

### Step 1: Differentiate \( y = (x + 5)^2 \)

1. **Identify the function**: Here, \( y = (x + 5)^2 \).
2. **Apply the power rule**: The derivative of \( x^n \) is \( n \cdot x^{n-1} \).
3. **Differentiate**:
\[ \frac{dy}{dx} = 2(x + 5)^{2-1} \cdot \frac{d}{dx}(x + 5) = 2(x + 5) \cdot 1 = 2(x + 5) \]

### Step 2: Differentiate \( y = 2e^3 \)

1. **Identify the function**: Here, \( y = 2e^3 \).
2. **Recognize that it is a constant**: Since there is no \( x \) term, this is a constant.
3. **Differentiate**:
\[ \frac{dy}{dx} = 0 \]

### Step 3: Differentiate \( y = (x + 5)^{-1/2} \)

1. **Identify the function**: Here, \( y = (x + 5)^{-1/2} \).
2. **Apply the power rule and chain rule**:
\[ \frac{dy}{dx} = -\frac{1}{2}(x + 5)^{-3/2} \cdot \frac{d}{dx}(x + 5) = -\frac{1}{2}(x + 5)^{-3/2} \cdot 1 = -\frac{1}{2}(x + 5)^{-3/2} \]

### Step 4: Differentiate \( y = 5x^{3/2} \)

1. **Identify the function**: Here, \( y = 5x^{3/2} \).
2. **Apply the power rule**:
\[ \frac{dy}{dx} = 5 \cdot \frac{3}{2} x^{3/2 - 1} = \frac{15}{2} x^{1/2} \]

### Final Answers

1. For \( y = (x + 5)^2 \): \( \frac{dy}{dx} = 2(x + 5) \)
2. For \( y = 2e^3 \): \( \frac{dy}{dx} = 0 \)
3. For \( y = (x + 5)^{-1/2} \): \( \frac{dy}{dx} = -\frac{1}{2}(x + 5)^{-3/2} \)
4. For \( y = 5x^{3/2} \): \( \frac{dy}{dx} = \frac{15}{2} x^{1/2} \)

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