Partial derivatives of multivariable functions in three or more variables
Partial derivatives of multivariable functions
Sometimes we need to find partial derivatives for functions with three or more variables, and we’ll do it the same way we found partial derivatives for functions in two variables.
We’ll take the derivative of the function with respect to each variable separately, which means we’ll end up with one partial derivative for each of our variables.
When we take the derivative with respect to one variable, we’ll treat all the other variables as constants.
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Finding partial derivatives for three- and four-variable functions
Example
Find the partial derivatives of the function.
???f(x,y,z)=3x^5y^2z^3???
With three variables ???x???, ???y???, and ???z???, we need to find three partial derivatives. When we take the partial derivative with respect to one variable, we’ll hold all others constant.
???\frac{\partial{f}}{\partial{x}}=3\left(5x^4\right)y^2z^3???
???\frac{\partial{f}}{\partial{x}}=15x^4y^2z^3???
and
???\frac{\partial{f}}{\partial{y}}=3x^5(2y)z^3???
???\frac{\partial{f}}{\partial{y}}=6x^5yz^3???
and
???\frac{\partial{f}}{\partial{z}}=3x^5y^2\left(3z^2\right)???
???\frac{\partial{f}}{\partial{z}}=9x^5y^2z^2???
???\partial f/\partial x??? is the partial derivative of the function ???f??? with respect to ???x???, ???\partial f/\partial y??? is the partial derivative of the function ???f??? with respect to ???y???, and ???\partial f/\partial z??? is the partial derivative of the function ???f??? with respect to ???z???.
Let’s try a more complex example with more than three variables.
We’ll take the derivative of the function with respect to each variable separately, which means we’ll end up with one partial derivative for each of our variables.
Example
Find the partial derivatives of the function.
???f(w,x,y,z)=2wx^4-\frac{3y^3}{7z^2}+\sin{2x}???
With four variables ???w???, ???x???, ???y???, and ???z???, we need to find four partial derivatives. When we take the partial derivative with respect to one variable, we’ll hold all others constant.
???\frac{\partial{f}}{\partial{w}}=2x^4???
and
???\frac{\partial{f}}{\partial{x}}=2w\left(4x^3\right)+(2)\cos{2x}???
???\frac{\partial{f}}{\partial{x}}=8wx^3+2\cos{2x}???
and
???\frac{\partial{f}}{\partial{y}}=-\frac{3\left(3y^2\right)}{7z^2}???
???\frac{\partial{f}}{\partial{y}}=-\frac{9y^2}{7z^2}???
and
???\frac{\partial{f}}{\partial{z}}=-\frac{3y^3}{7}(-2)z^{-3}???
???\frac{\partial{f}}{\partial{z}}=\frac{6y^3}{7z^3}???
???\partial f/\partial w??? is the partial derivative of the function ???f??? with respect to ???w???, ???\partial f/\partial x??? is the partial derivative of the function ???f??? with respect to ???x???, ???\partial f/\partial y??? is the partial derivative of the function ???f??? with respect to ???y???, and ???\partial f/\partial z??? is the partial derivative of the function ???f??? with respect to ???z???.
