First slide

Derivatives

Question

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

Moderate

A

12
B

32
C

10
D

0

Solution

$\frac{d x}{d s} = 1, \frac{d y}{d s} = 2 ....... (1)$
$\frac{d^{2} x}{d s^{2}} = 0, \frac{d^{2} y}{d s^{2}} = 0 ....... (2)$
$u = x^{2} + y^{2}$
$\begin{array}{l} \frac{d u}{d s} = 2 x \frac{d x}{d s} + 2 y \frac{d y}{d s} \\ \frac{d^{2} u}{d s^{2}} = 2 {(\frac{d x}{d s})}^{2} + 2 x \frac{d^{2} x}{d s^{2}} + 2 {(\frac{d y}{d s})}^{2} + 2 y (\frac{d^{2} y}{d s^{2}}) \end{array}$
$\begin{matrix} From 1 and 2 \frac{d^{2} u}{d s^{2}} = 2 \times 1 + 0 + 2 \times 4 + 0 \\ = 2 + 8 = 10 \end{matrix}$

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