Finding the gradient of an expression

Question

I'm trying to solve the following problem:

Let $\textbf{r} = (x,y) = x\textbf{i} + y\textbf{j}$ and $r = ||\textbf{r}|| = \sqrt{x^2+y^2}$. Also $\textbf{r} \neq 0$.

Let $f(x,y) = r^m$. what is the correct expression for $\bigtriangledown$f?

I'm thinking $mr^{(m-2)}\textbf{r}$. what do you guys think?

Answer 1

$$f(x,y) = r^m = \left(\sqrt{x^2 + y^2}\right)^{m}$$

$$\implies \nabla f = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} $$

$$\implies \nabla f = m\left(\sqrt{x^2 + y^2}\right)^{m-1} \cdot \frac{2x}{2\sqrt{x^2+y^2}}\hat{i} + m\left(\sqrt{x^2 + y^2}\right)^{m-1} \cdot \frac{2y}{2\sqrt{x^2+y^2}}\hat{j}$$

$$\implies \nabla f = mr^{m-2}\textbf{r}$$

Answer 2

Worked explicitly: $$\frac\partial{\partial x}(x^2+y^2)^{m/2}=2x(m/2)(x^2+y^2)^{(m-2)/2}=mxr^{m-2}$$ Similarly, the partial derivative on $y$ is $myr^{m-2}$. You are correct.