Question

If \(u = x^2 + y^2\) and \(x = s + 3t\), \(y = 2s - t\) then \(\frac{d^2 u}{ds^2}\) is equal to

• A. 12
• B. 10
• C. 32
• D. 36

Hint:

Treat \(t\) as constant when differentiating with respect to \(s\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Express \(u\) in terms of \(s\) and \(t\), then differentiate.
Step 2: Detailed Explanation:
\(u = (s + 3t)^2 + (2s - t)^2 = s^2 + 6st + 9t^2 + 4s^2 - 4st + t^2\)
= \(5s^2 + 2st + 10t^2\)
\(\frac{du}{ds} = 10s + 2t\)
\(\frac{d^2 u}{ds^2} = 10\)
Step 3: Final Answer:
\(\frac{d^2 u}{ds^2} = 10\).