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.

##### Examples

.

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.