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

To find the second derivative \(\frac{d^2 u}{ds^2}\) given that \(u = x^2 + y^2\) and the substitutions \(x = s + 3t\) and \(y = 2s - t\), we start by expressing \(u\) in terms of \(s\) and \(t\).

First compute \(x^2 + y^2\):

1. Substitute \(x\) and \(y\): \(x = s + 3t\) and \(y = 2s - t\).
2. Calculate \(x^2\): \(x^2 = (s + 3t)^2 = s^2 + 6st + 9t^2\).
3. Calculate \(y^2\): \(y^2 = (2s - t)^2 = 4s^2 - 4st + t^2\).
4. Now, express \(u\) using: \(u = x^2 + y^2 = s^2 + 6st + 9t^2 + 4s^2 - 4st + t^2\)

Combine and simplify the expressions:

• Combine like terms: \(u = 5s^2 + 2st + 10t^2\).

Now differentiate \(u\) with respect to \(s\):

1. Find the first derivative \(\frac{du}{ds}\): \(\frac{du}{ds} = \frac{d}{ds}(5s^2 + 2st + 10t^2) = 10s + 2t\).
2. Find the second derivative: \(\frac{d^2u}{ds^2} = \frac{d}{ds}(10s + 2t) = 10\).

Thus, the second derivative \(\frac{d^2 u}{ds^2}\) is \(10\), which matches the given correct answer.