Partial derivatives, rough introduction

Summary

\(f\) is a real-valued function of two real variables , \(c\)
The partial derivative of \(f\) with respect to \(x\) , denoted by \(f'_x\left( x,y \right)\) , can be calculated as the ordinary derivative of \(f\) as a function of \(x\) while temporarily treating \(y\) as a constant.
The partial derivative of \(f\) with respect to \(y\) , denoted by \(f'_y\left( x,y \right)\) , can be calculated as the ordinary derivative of \(f\) as a function of \(y\) while temporarily treating \(x\) as a constant.
Example: \(f\left( x,y \right)=x^2y^2\)

\[ \frac{∂f\left( x,y \right)}{∂x} \; \mathrm{or} \; \frac{∂}{∂x}f\left( x,y \right) \; \mathrm{or} \; \frac{∂z}{∂x}\]

\[ \frac{∂f\left( x,y \right)}{∂y} \; \mathrm{or} \; \frac{∂}{∂y}f\left( x,y \right) \; \mathrm{or} \; \frac{∂z}{∂y}\]

\(z=f\left( x,y \right)\) and \(y\) is constant ( \(Δy=0\) ). If \(x\) increases by a small value \(Δx\) then \(z\) will increase by approximately

\[Δz≈f'_x\left( x,y \right)Δx\]

\(f'_x\left( x,y \right)\) is the approximate increase in \(z\) if \(x\) increases by 1 unit while \(y\) is held constant.
\(z=f\left( x,y \right)\) and \(x\) is constant ( \(Δx=0\) ). If \(y\) increases by a small value \(Δy\) then \(z\) will increase by approximately

\[Δz≈f'_y\left( x,y \right)Δy\]

\(f'_y\left( x,y \right)\) is the approximate increase in \(z\) if \(y\) increases by 1 unit while \(x\) is held constant.
\(z=f\left( x,y \right)\) . If \(x\) increases by a small value \(Δx\) and \(y\) increases by a small value \(Δy\) then \(z\) will increase by approximately

\[Δz≈f'_x\left( x,y \right)Δx+f'_y\left( x,y \right)Δy\]

The partial derivative of \(f'_x\left( x,y \right)\) with respect to \(x\) is denoted by \(f_{xx}^{''}\left( x,y \right)\) or

\[ \frac{∂^2f\left( x,y \right)}{∂x^2}\]

The partial derivative of \(f'_x\left( x,y \right)\) with respect to \(y\) is denoted by \(f_{xy}^{''}\left( x,y \right)\) or

\[ \frac{∂^2f\left( x,y \right)}{∂x∂y}\]

The partial derivative of \(f'_y\left( x,y \right)\) with respect to \(x\) is denoted by \(f_{yx}^{''}\left( x,y \right)\) or

\[ \frac{∂^2f\left( x,y \right)}{∂y∂x}\]

The partial derivative of \(f'_y\left( x,y \right)\) with respect to \(y\) is denoted by \(f_{yy}^{''}\left( x,y \right)\) or

\[ \frac{∂^2f\left( x,y \right)}{∂y^2}\]

If \(f\) is a smooth function then the mixed partial derivatives are equal, \(f_{xy}^{''}\left( x,y \right)=f_{yx}^{''}\left( x,y \right)\) , or,

\[ \frac{∂^2f\left( x,y \right)}{∂x∂y}= \frac{∂^2f\left( x,y \right)}{∂y∂x}\]

This is called the Young's theorem (or the Clairaut's theorem or the equality of mixed partials).