If ` u = x^(2) + y^(2) " and " x = s + 3t, y = 2s - t , " then " (d^(2) u)/(ds^(2))` is equal to

Options

A. 12
B. 32
C. 36
D. 10

Answer

To solve the problem, we need to find the second derivative of \( u \) with respect to \( s \), where \( u = x^2 + y^2 \), \( x = s + 3t \), and \( y = 2s - t \).

### Step-by-Step Solution:

1. **Express \( u \) in terms of \( s \)**:
\[ u = x^2 + y^2 = (s + 3t)^2 + (2s - t)^2 \]

2. **Expand \( u \)**:
\[ u = (s^2 + 6st + 9t^2) + (4s^2 - 4st + t^2) \]
\[ u = s^2 + 6st + 9t^2 + 4s^2 - 4st + t^2 \]
\[ u = (1 + 4)s^2 + (6 - 4)st + (9 + 1)t^2 \]
\[ u = 5s^2 + 2st + 10t^2 \]

3. **Differentiate \( u \) with respect to \( s \)**:
\[ \frac{du}{ds} = \frac{d}{ds}(5s^2 + 2st + 10t^2) \]
\[ \frac{du}{ds} = 10s + 2t \]

4. **Differentiate again to find \( \frac{d^2u}{ds^2} \)**:
\[ \frac{d^2u}{ds^2} = \frac{d}{ds}(10s + 2t) \]
Since \( t \) is treated as a constant with respect to \( s \):
\[ \frac{d^2u}{ds^2} = 10 \]

### Final Answer:
\[ \frac{d^2u}{ds^2} = 10 \]