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

For functions with multiple independent variables f(x,y), f(x,y,z), etc. when you take a derivative you need to specify which variable you’re differentiating the function by, so basically which independent variable you’re using to measure the function’s rate of change.

So for a function f(x,y) you have partial derivatives df/dx and df/dy. For function f(x,y,z) partial derivatives df/dx, df/dy, df/dz and so on.

You can imagine for a function f(x,y) say the function is entirely flat when moving along the x-axis, no change in f(x,y) whatsoever, in which case df/dx = 0 saying there’s no change in f(x,y) with respect to x. Conversely say there’s a great amount of exponential change in the function as you move along the y-axis so you’re partial derivative there could be df/dy = e^y . This is just to highlight that partial derivatives for different variables can be drastically different depending on the function.