Partial derivatives

Summary

Functions of two variables

\(f\) is a function of two variables and \(\left( x,y \right)\) is in the domain of \(f\) .

\[f'_x\left( x,y \right)=\lim_{h→0} \frac{f\left( x+h,y \right)-f\left( x,y \right)}{h}\]

is called the partial derivative of \(f\) with respect to \(x\) at \(\left( x,y \right)\) (assuming that the limit exists)
Similarly,

\[f'_y\left( x,y \right)=\lim_{h→0} \frac{f\left( x,y+h \right)-f\left( x,y \right)}{h}\]

is called the partial derivative of \(f\) with respect to \(y\) at \(\left( x,y \right)\) (assuming that the limit exists)
If both partial derivatives \(f'_x\left( x,y \right)\) and \(f'_y\left( x,y \right)\) exists and are continuous close to \((x,y)\) then \(f\) is differentiable at \((x,y)\) . (This is not the formal definition of differentiability but a sufficient condition that is easier to check).

Functions of \(n\) variables

If \(f\) is a function of \(n\) variables and if \(x_0=\left( x_1,⋯,x_n \right)\) is in the domain of \(f\) and the limit

\[\lim_{h→0} \frac{f\left( x_1,⋯,x_i+h,⋯,x_n \right)-f\left( x_1,⋯,x_n \right)}{h}\]

exists then the limit is called the partial derivative of \(f\) with respect to \(x_i\) at \(x\) denoted by \(f'_i(x)\) .
If \(f\) is a function of \(n\) variables then the \(n×1\) column vector

\[\begin{pmatrix} f'_1(x) \\ f'_2(x) \\ \vdots \\ f'_n(x) \end{pmatrix}\]