# 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???

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## Finding the differential of a multivariable function

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}???

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

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.

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