What exactly is a partial derivative?

Question

What exactly is a partial derivative? I have a basic idea of a derivative i.e, I know that dy/dx is rate of change of y with respect to x and that it represents the slope of a curve

Thanks!

In: Mathematics

Answer 1

ELI5 example: Nickels are worth 5 cents and quarters are worth 25 cents. So if you have n nickels and q quarters, you have a total value of 5*n + 25*q cents, or v(n,q) = 5n + 25q.

The partial derivative of v (with respect to one of the parameters) tells you how much it increases for an increase in that one parameter. There’s a different partial derivative for each parameter. Here we have 2 (number of nickels and number of quarters), so there are two partial derivatives:

If you add one more nickel, the value goes up 5 cents, so the partial derivative of v(n,q) with respect to n is 5 (or dv/dn = 5). Similarly, adding one quarter gives 25 more cents, so dv/dq = 25.