Partial Derivative

A derivative used as a part of a larger multidimensional derivative. We only derive one term and treat all others as constants.

Notations

• $\frac{\partial }{\partial x}f(x,y)$
• $f_x$

Example

Find the derivative $g(x,y)=x^2+y^2+1$

Soln

Fix variable $y$ and consider the rate of change over variable $x$ $⟹\frac{\partial f}{\partial x}=\frac{\partial }{\partial x}(x^2+y^2+1)=2x$ Fix variable $x$ and consider the rate of change over variable $y$ $⟹\frac{\partial g}{\partial y}=\frac{\partial }{\partial y}(x^2+y^2+1)=2y$ $⟹$ derivative $g^{\prime }(x,y)=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})=(2x,2y)$