Let $f(x_1,x_2)=\ln(1+2x_1+4x_2+x_1x_2),\vec{x}=(x_1,x_2)\in\Bbb{R}^2$ , show that $f$ is continuously differentiable at $\vec{x}=\vec{0}$.
Should I first prove $f$ is differentiable then prove its derivatives is continuous? I am newly to multivariable calculus, what is the derivative of this kind of function exactly?
In any neighborhood of $(0,)$ where $|2x_1+4x_2+x_1x_2| <1$ the function has continuous partial derivatives. Hence $f$ is continuously differentiable in it. In fact, $f$ is infinitely differentiable in a neighborhood of $(0,0)$. I particular this if true on the ball of radius $\frac 1 7$ around $(0,0)$.