I was reading in my book that if we have $f(x,y)=0$ and want to find $\frac{dy}{dx}$ with respect to $x$ we use the formula $y'_x=-\cfrac{f'_x}{f'_y}$. I am not very familiar with this formula so I have these questions:
Question $1$:How we can prove this formula works?
Question$2$:Is it really necessary to memorize this formula to find $\frac{dy}{dx}$ ? I want to know your experience in finding derivative in this form. I think it is easier to differentiate both sides of the equation instead of memorizing some formula. so is it really useful formula?(do you recommend to ignore this formula or use it?)
There are two ways to pull out this formula. One is using the 'implicit function theorem' and the second is using the concept of 'gradient'. The first requires a little multivariable calculus and the second requires a little vector calculus.
For the gradient way, see here
For the multivariable way (implicit function theorem), see here