Question

If $u=x^2+y^2$ and $x=s+3t,y=2s-t,$ where $s$ and $t$ are independent of each other, then the value of $\frac{d^{2}u}{d{s}^2}$ is

Answer 1

Given : $u=x^2+y^2$
$\Rightarrow \frac{du}{ds}=2x\cdot \frac{dx}{ds}+2y\cdot \frac{dy}{ds}$
Now,
$x=s+3t\Rightarrow \frac{dx}{ds}=1y=2s-t\Rightarrow \frac{dy}{ds}=2$
Then
$\Rightarrow \frac{du}{ds}=2x+4y\Rightarrow \frac{d^{2}u}{d{s}^2}=2\frac{dx}{ds}+4\frac{dy}{ds}\Rightarrow \frac{d^{2}u}{d{s}^2}=2+8=10$