Definition of partial differential equation with example?

Answer 1

A partial derivative is the derivative in respect to one dimension. You can use the rules and tricks of normal differentiation with partial derivatives if you hold the other variables as constants, but the actual definition is very similar to the definition of a normal derivative. In respect to x, it looks like:

fx(x,y)=[f(x+Δx,y)-f(x,y)]/Δx

and in respect to y:

fy(x,y)=[f(x,y+Δy)-f(x,y)]/Δy

Here's an example. take the function z=3x2+2y

we want to find the partial derivative in respect to x, so we can use basic differentiation techniques if we treat y as a constant, so zx'=6x+0 because the derivative of a constant (2y in this case) is always 0. this applies to any number of dimensions. if you were finding the partial in respect to a of f(a,b,c,d,e,f,g), you would just differentiate as normal and hold b through g as constants.