Continuous partial derivatives

Question

I have the following function and I want to show that it is differentiable. I am going to do this by showing that the partial derivatives are continuous and so I will show that they are continuous at (0,0). So, i am going to show that as the limit of (x,y) approaches (0,0) the derivative approaches 0.

$$f(x,y)=\left\{\begin{array}{l} \frac{x^2y^2}{\sqrt{x^2+y^2}},\:\text{if $(x,y) \not= (0,0)$;}\\ 0,\:\text{if $(x,y)=(0,0)$;} \end{array}\right.$$

$$\begin{cases} \dfrac {\partial f}{\partial x}(x,y)= \dfrac{x^3y^2+2xy^4}{({x^2+y^2})^\frac{3}{2}}\\ \dfrac {\partial f}{\partial x}(0,0)=0\end{cases}$$

I am having trouble however and was wondering if anyone could work me through this case.

Answer 1

We have

$$\left|\dfrac{x^3y^2+2xy^4}{{(x^2+y^2)^{3/2}}}\right|\le \dfrac{|x|(x^2+y^2)^2+2|x|(x^2+y^2)^2}{(x^2+y^2)^{3/2}}=3|x|(x^2+y^2)^{1/2}\xrightarrow{(x,y)\to(0,0)}0$$ so $\dfrac {\partial f}{\partial x}(x,y)$ is continuous at $(0,0)$

Answer 2

We have $|x|\le\sqrt{x^2+y^2}$ and $|y|\le\sqrt{x^2+y^2}$. Thus, the absolute value of the numerator does not exceed $3(x^2+y^2)^{5/2}$ and the fraction does not exceed $3(x^2+y^2)$ in absolute value and hence tends to zero as $(x,y)\to(0,0)$