The fastest variation of temperature will be given as $$\frac{\partial \phi}{\partial x}i+\frac{\partial\phi }{\partial y}j$$ .

Here $\phi$ is any function in terms of $x$ and $y$.

$$\frac{\partial \phi}{\partial x}=\frac{\partial }{\partial x}\left ( x^{3}+4xy\right ) \\=3x^{2}+4y$$

$$\frac{\partial \phi }{\partial y}=\frac{\partial }{\partial y}\left ( x^{3} +4xy\right ) \\=4x$$

At point $(1,0)$ its value will be $$3x^{2}+4y=3(1^{2})+4(0)=3$$

and $$4x=4(1)=4$$

$$=3i+4j$$

At this point its directional unit vector will be $$\frac{3i+4j}{\sqrt{3^{2}+4^{2}}} \\=\frac{3i+4j}{\sqrt{25}} \\=\frac{3i+4j}{5}$$

$$=0.6i+0.8j$$