Cosine of the angle of intersection of curves `y = 3^(x-1) logx and y= x^(x)-1` at (1,0) is

Options

A. 1
B. `1/2`
C. 0
D. `1/3`

Answer

To find the cosine of the angle of intersection of the curves \( y = 3^{(x-1)} \log x \) and \( y = x^x - 1 \) at the point \( (1, 0) \), we will follow these steps:

### Step 1: Find the derivatives of the curves

1. **For the first curve \( y_1 = 3^{(x-1)} \log x \)**:
- We can rewrite \( y_1 \) as \( y_1 = e^{(x-1) \ln 3} \log x \).
- To differentiate, we will use the product rule:
\[ \frac{dy_1}{dx} = \frac{d}{dx}(e^{(x-1) \ln 3}) \cdot \log x + e^{(x-1) \ln 3} \cdot \frac{d}{dx}(\log x) \]
- The derivative of \( e^{(x-1) \ln 3} \) is \( \ln 3 \cdot e^{(x-1) \ln 3} \).
- The derivative of \( \log x \) is \( \frac{1}{x} \).
- Thus, we have:
\[ \frac{dy_1}{dx} = \ln 3 \cdot 3^{(x-1)} \log x + 3^{(x-1)} \cdot \frac{1}{x} \]

2. **For the second curve \( y_2 = x^x - 1 \)**:
- We can differentiate \( y_2 \) using the formula for \( x^x \):
\[ \frac{dy_2}{dx} = x^x \left( \ln x + 1 \right) \]

### Step 2: Evaluate the derivatives at the point \( (1, 0) \)

1. **Evaluate \( \frac{dy_1}{dx} \) at \( x = 1 \)**:
\[ \frac{dy_1}{dx} \bigg|_{x=1} = \ln 3 \cdot 3^{(1-1)} \log 1 + 3^{(1-1)} \cdot \frac{1}{1} = \ln 3 \cdot 1 \cdot 0 + 1 \cdot 1 = 1 \]

2. **Evaluate \( \frac{dy_2}{dx} \) at \( x = 1 \)**:
\[ \frac{dy_2}{dx} \bigg|_{x=1} = 1^1 \left( \ln 1 + 1 \right) = 1 \cdot (0 + 1) = 1 \]

### Step 3: Find the angle of intersection

- Let \( m_1 = \frac{dy_1}{dx} \bigg|_{x=1} = 1 \) and \( m_2 = \frac{dy_2}{dx} \bigg|_{x=1} = 1 \).
- The formula for the tangent of the angle \( \theta \) between two curves is given by:
\[ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]
- Substituting the values:
\[ \tan \theta = \left| \frac{1 - 1}{1 + 1 \cdot 1} \right| = \left| \frac{0}{2} \right| = 0 \]
- Therefore, \( \theta = \tan^{-1}(0) = 0 \).

### Step 4: Find the cosine of the angle

- The cosine of the angle \( \theta \) is:
\[ \cos \theta = \cos(0) = 1 \]

### Final Answer

The cosine of the angle of intersection of the curves at the point \( (1, 0) \) is \( \boxed{1} \).