The second partial derivative for a function with two or more variables is found by differentiating twice in terms of one variable or in terms of one variable and then another variable. For instance, a function with the variables x and y has three second partial derivatives, one in terms of x, one in terms of y and one in terms of mixed variables.

For the function f(x, y)=(x^2)(y^3), the second partial derivative in terms of x is found by differentiating once in terms of x, and then differentiating the resulting function again in terms of x. The first partial derivative with respect to x is 2x(y^3), as the derivative of x^2 is 2x. The second partial derivative with respect to x is 2(y^3). Again, this derivative is found by differentiating 2x. For the same function, the second partial derivative can also be obtained with respect to the variable y. The first partial derivative with respect to y is 3(x^2)(y^2), while the second partial derivative with respect to y is 6(x^2)y. The second partial derivative of any function with multiple variables can be found using this technique. The first partial derivative with respect to x is written as f_x(x, y), while the second partial derivative with respect to x is written as f_xx(x, y). To find a mixed partial derivative, the function is differentiated first in terms of one variable and then in terms of the other. Regardless of order, the result is the same. For example, the mixed partial derivative of the above function is 6x(y^2).