Partial derivative

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In multivariable calculus, the partial derivative of a function is the derivative of one variable in the function while supposing all other variables to act as constants whose derivative is zero. In other words, a partial derivative essentially holds all variables constant except the one being differentiated. Partial derivatives are often used in multivariable functions.

For partial derivatives of function f with respect to variable x, the notation

${\displaystyle {\frac {\partial f}{\partial x}}}$, ${\displaystyle f_{x}}$, ${\displaystyle \partial _{x}f}$

is standard, but other notations are sometimes used.

Examples

If we have a function ${\displaystyle f(x,y)=x^{2}+y}$, then there are several partial derivatives of f(x, y) that are all equally valid. For example,

${\displaystyle {\frac {\partial }{\partial y}}[f(x,y)]=1}$ {In the partial derivative of y, x is ignored/considered a constant whose derivative is zero}

Or, we can do the following:

${\displaystyle {\frac {\partial }{\partial x}}[f(x,y)]=2x}$ {In the partial derivative of x, y is ignored/considered a constant whose derivative is zero}

Related pages

• Difference quotient
• Partial differential equation

References

1. ↑ "List of Calculus and Analysis Symbols". Math Vault. 2020-05-11. Retrieved 2020-09-16.
2. ↑ Weisstein, Eric W. "Partial Derivative". mathworld.wolfram.com. Retrieved 2020-09-16.
3. ↑ "Calculus III - Partial Derivatives". tutorial.math.lamar.edu. Retrieved 2020-09-16.