Question

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

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

Answer 1

Answer: (D)

Given, $u=x^2+y^2$,
$x=s+3t$,
$y=2s-t$
$\Rightarrow \frac{dx}{ds}=1,\frac{dy}{ds}=2$
Now, $u=x^2+y^2$
$\Rightarrow \frac{du}{ds}=2x\frac{dx}{ds}+2y\frac{dy}{ds}$
$=2x+4y$
$\frac{d^{2}u}{d{s}^2}=2\left(\frac{dx}{ds}\right)+4\left(\frac{dy}{ds}\right)$
$\Rightarrow \frac{d^{2}u}{d{s}^2}=2\left(1\right)+4\left(2\right)$
$=10$