Gradient of $f(x,y)=\sqrt{x^2(1+y)+y^2+4}$

Question

I have to solve the gradient of $f(x,y)=\sqrt{x^2(1+y)+y^2+4}$ as a part of a larger task. I know how to do this with partial derivatives but I was wondering if there are simpler ways to find the gradient since the nature of $f$ makes partial derivation a little messy?

Answer 1

Hint: $$\frac{\partial f(x,y)}{\partial x}=\frac{1}{2}\left(x^2(1+y)+y^2+4\right)^{-1/2}2x(1+y)$$

Answer 2

You could use some cunning notation, for example:

$$\frac{d f}{d x}=\frac{x(1+y)}f$$

and etc.