How to calculate the differential of any multivariable function

Formulas for the differential of a multivariable function

The differential of a multivariable function is given by

???dz=\frac{\partial{z}}{\partial{x}}\ dx+\frac{\partial{z}}{\partial{y}}\ dy???

???\frac{\partial{z}}{\partial{x}}??? is the partial derivative of ???f??? with respect to ???x???

???\frac{\partial{z}}{\partial{y}}??? is the partial derivative of ???f??? with respect to ???y???

Example

Find the differential of the multivariable function.

???z=6x^2y-4\ln{y}???

Before we can use the formula for the differential, we need to find the partial derivatives of the function with respect to each variable.

???\frac{\partial{z}}{\partial{x}}=6(2x)y???

???\frac{\partial{z}}{\partial{x}}=12xy???

and

???\frac{\partial{z}}{\partial{y}}=6x^2-4\left(\frac{1}{y}\right)???

???\frac{\partial{z}}{\partial{y}}=6x^2-\frac{4}{y}???

We’ll plug the partial derivatives into the formula for the differential.

???dz=(12xy)dx+\left(6x^2-\frac{4}{y}\right)dy???

???dz=12xy\ dx+6x^2\ dy-\frac{4}{y}\ dy???

This is the differential of the function.