$\mathrm{If} \mathrm{u}=\mathrm{f}(\mathrm{x}+\mathrm{ay})+\mathrm{g}(\mathrm{x}-\mathrm{ay}) \mathrm{then} \frac{{\partial }^2\mathrm{u}}{ \partial {\mathrm{y}}^2 } \mathrm{is}$ _____.

The minimum value of ${\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2$ with the condition $\mathrm{xy}+\mathrm{yz}+\mathrm{zx}=3{\mathrm{a}}^2$ is _____.

The directional derivative of $2{\mathrm{x}}^2+3{\mathrm{y}}^2+{\mathrm{z}}^2$ at the point $\mathrm{P}(2,1,3)$ in the direction of the vector $\hat{\mathrm{i}}-2\hat{\mathrm{k}}$ is _____.

If ${\mathrm{x}}^2-{\mathrm{y}}^2+\sin \left(\mathrm{xy}\right)=0$ , then $\frac{\mathrm{dy}}{ \mathrm{dx} }=$ _____.

Let $\overline{\mathrm{r}}=\mathrm{x}\hat{\mathrm{i}}+\mathrm{y}\hat{\mathrm{j}}+\mathrm{z}\hat{\mathrm{k}}$ and $\mathrm{r}=\left|\overline{\mathrm{r}}\right|$ , then $\nabla {\mathrm{e}}^{\mathrm{r}}=$ _____.

The directional derivative of $2{\mathrm{x}}^2+3{\mathrm{y}}^2+{\mathrm{z}}^2$at the point $\mathrm{P}(2,1,3)$ in the direction of the vector $\hat{\mathrm{i}}+2\hat{\mathrm{j}}$ is _____.

Find $\frac{d\mathrm{u}}{d\mathrm{t}}$ , if $\mathrm{u}=4\mathrm{x}-{\mathrm{y}}^2, \mathrm{x}=\cos \left(2\mathrm{t}\right) \& \mathrm{y}=\sin \left(2\mathrm{t}\right).$

The function ${\mathrm{x}}^2+{\mathrm{y}}^2+6\mathrm{x}+10$ has minimum value at point _____.

The value of the limit $\underset{(\mathrm{x}, \mathrm{y})\to (0, 0)}{\lim }\frac{3{\mathrm{x}}^2\mathrm{y}}{ {\mathrm{x}}^2+{\mathrm{y}}^2 }$ is _____.

If ${\mathrm{x}}^2-{\mathrm{y}}^2-\sin \left(\mathrm{xy}\right)=0$ , then $\frac{\mathrm{dy}}{ \mathrm{dx} }=$ _____.