Derivative of the function $f(x,y)=\frac{x^2y}{x^2+y^2}$ if $(x,y)≠(0,0)$.

Question

I'm working on this calculus problem but I'm really stuck, any help would be appreciated

Let the function $f:ℝ^2 →ℝ$ be defined as:

$f(x,y)=\dfrac{(x^2y)}{x^2+y^2}$ if $(x,y)≠(0,0)$

$f(x,y)=0$ if $(x,y)=(0,0)$

(a) Use the definition of partial derivatives to find $f_x(0,0)$ and $f_y(0,0)$

I have solved this using limits and arrived at the answer that both partial derivatives at $(0,0)$ are zero.

(b)Let $a∈ℝ$ be a non-zero constant and let $r(t)=(t,at)$. Show that $f◦r$ is differentiable and compute $D(f◦r)$

Here I really don't know how to start, I don't how to compute $f◦r$

I think once I the proper function and a general idea of how to approach this problem I'll be able to solve it myself

Thanks in advance

Answer 1

$f\circ r$ is just the map $f$ with $t$ instead of $x$'s and $at$ instead of $y$'s. I.e. $$(f\circ r)(t)=\frac{t^2(at)}{t^2+(at)^2} = \frac{at}{1+a^2}$$ Notice that this is just a scalar function so the derivative is the regular Calculus I derivative.