Show that multivariate function is differentiable

Question

Given the function $$f(x,y)=x^2-4x+y^2$$

$\textbf{Show that f is differentiable}$

I know that a multivariate function if differentiable if it has partial derivatives in an open area A and if the partial derivatives are continuous in A.

I found the partial derivatives to be $$\frac{df}{dx}=2x-4 \text{ and } \frac{df}{dy}=2y$$

How can I get further from here?

Answer 1

$$\begin{align}f(x,y)&=x^2-4x+y^2\\f_x&=2x-4,f_y=2y\\f_{xy}&=2=f_{yx}\end{align}$$

Since $\boxed{f_{xy}=f_{yx}}$ hence $f(x,y)$ is differentiable on $\mathbb R$

Answer 2

not that $$x^2-4x$$ and $$y^2$$ are differentiable thus the sum $$x^2-4x+y^2$$ is differentiable