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

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Solution

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

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If u=x2+y2 and x=s+3t, y=2s−t, where s and t are independent of each other, then the value of d2uds2 is

Given : u=x2+y2 ⇒duds=2x⋅dxds+2y⋅dyds Now, x=s+3t⇒dxds=1y=2s−t⇒dyds=2 Then ⇒duds=2x+4y⇒d2uds2=2dxds+4dyds⇒d2uds2=2+8=10

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