GATE (TF) Textile 2021 Question Paper Solution | GATE/2021/TF/01

Question (1): Let the function f(x,y) be defined as

$f(x,y)=\begin{cases} \frac{y}{|y|}\sqrt{2x^2+3y^2}, \quad y\neq 0 \\ 0, \quad \quad \quad \quad \quad \quad \quad y=0 \end{cases}$

Then $\frac{\partial f}{\partial y}(0,0)$ is equal to

[Show Answer]

$f (x,y) = \pm \sqrt{2x^2 + 3y^2}$

$\frac{\partial f}{\partial y} (x,y) = \frac{1}{2}\frac{1}{\sqrt{2x^2+3y^2}}\times (0+6y)$

$\frac{\partial f}{\partial y} (x,y) = \frac{3y}{\sqrt{2x^2+3y^2}}$

$\frac{\partial f}{\partial y} (x,y) = \frac{3}{\sqrt{2(\frac{x}{y})^2+3}}$

$\frac{\partial f}{\partial y} (0,0) = \frac{3}{\sqrt{2 \times 0 + 3}}$

$\frac{\partial f}{\partial y} (0,0) = \frac{3}{\sqrt{3}}$

$\frac{\partial f}{\partial y} (0,0) = \sqrt{3}$