at which points the function is differentiable

Question

Could anyone help me how to find those points in which $f(x,y)={x^3y\over x^4+y^2}; (x,y)\ne (0,0)$ is differentiable? is there any general formuale for calculationg directional derivative of $f$?

thanks for helping!

Answer 1

I would say it's differentiable everywhere where $(x,y)\ne(0,0)$. To see that one simply calculate partial derivates and realize that these are continuous everywhere except at $(0,0)$ - because the partial derivate is a polynomial divided by the denominator squared (as per the formula for derivate of a ratio).

The formula for directional derivate is $f'_v = \nabla f\cdot v$ where $\nabla f$ is the gradient (and $v$ is a vector of unit length). The gradient in turn is $\nabla f = (f'_x, f'_y)$ if $f$ is differentiable. If on the other hand $f$ is not differentiable there's no such general formula.