Partial Derivatives

To explain the concept of partial derivatives, let's look at a double-variable function, say f(x,y) = x² + xy + 2y³. To find a value for f(x,y), we take a point on the plane and plug its x and y values into the function where appropriate.

Partial derivatives behave pretty much like regular derivatives for single-variable functions. You take a partial derivative with respect to one of the function's variables - in this case, x or y. The other varibles behave like constants. In the example above, then, the partial derivative of f(x,y) with respect to x is
2x + y, since y behaves like a constant for the purposes of derivation.This can also be written f_x. The partial derivative of f(x,y) with respect to y, f_y, is x + 6y².

Really, that's all there is to it. For practise, try finding the partial derivatives with respect to x in each case below (the answers can be found here).

1) f(x,y) = x + y

2) g(x,y) = x³-5 + xy²

3) h(x,y) = sin(x)cosy(y)e^(sin(y)+53)