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???
Step-by-step example of how to calculate the differential
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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}???
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.