Question

If $u=x^2+y^2$ and $x=s+3t$, $y=2s-t$, where $t$ does not depend on $s$, then $\frac{d^{2}u}{d{s}^2}$ is

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

Answer 1

The correct option is D $10$

$u=x^2+y^2$, $x=s+3t$, $y=2s-t$
Differentiating $x$ and $y$ w.r. to $s$, we get
Now, $\frac{dx}{ds}=1$, $\frac{dy}{ds}=2$ ..............(1)
Again differentiating w.r. to $s$,
$\frac{d^{2}x}{d{s}^2}=0$, $\frac{d^{2}y}{d{s}^2}=0$ ..................(2)
Now, $u=x^2+y^2$,
Differentiate $u$ w.r. to $s$
$\frac{du}{ds}=2x\frac{dx}{ds}+2y\frac{dy}{ds}$
By taking double derivative, we get
$\frac{d^{2}u}{d{s}^2}=2{\left(\frac{dx}{ds}\right)}^2+2x\frac{d^{2}x}{d{s}^2}+2{\left(\frac{dy}{ds}\right)}^2+2y\left(\frac{d^{2}y}{d{s}^2}\right)$
Now from (1) and (2),
$\frac{d^{2}u}{d{s}^2}=2\times 1+0+2\times 4+0=10.$