Partial Derivatives 101

I realized some of you may not know what a partial derivative is, so here's the crash course in it. It is REALLY easy I promise you, just give me 30 seconds:

So we all know that the derivative of x = 1, and of 4x = 4

Usually, you are used to implicit differentiation when you see a 'y', that is putting a 'dy/dx' in for the derivative of 'y'. This is not the case with partial derivatives

When you take the partial derivative of something with respect to x, treat any 'y' in the function as if it were a constant, and use all appropriate rules (Chain, product, quotient, etc.) as normal.

So the derivative of z = xy with respect to x (a.k.a. dz/dx) = x'y + y'x (Product Rule).
*Note that since y is a constant, the derivative of y = 0, y'x is actually 0. The derivative of x is just 1.

Therefore, the derivative of z = xy is y'x + x'y which is 0+1y

Thus: dz/dx = y

By the same logic, dz/dy (partial derivative with respect to y) = x

Good luck!