Consider the curve in which the surface meets the plane a constant.
In this plane , and would be a formula for the gradient of the tangent to the curve.
If we differentiate with respect to x, treating the y as if it was a constant (some say holding y constant), the the derivative obtained is called the partial derivative of with respect to x and we write or or . Similarily we have .
With functions of more than two variables one differentiates with respect to one of the variables by holding all the other variables constant.
In the obvious way we can have higher order partial derivatives.
Similarly for we have:
For commonly encountered functions we have that
From here on we may assume that all the mixed derivatives are equal.
Note, the normal rules (sum, product, quotient, function of a function) of differentiation apply to partial differentiation.