# 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: $$f\left( x,y \right)=x^2y^2$$
• $$f'_x\left( x,y \right)=2xy^2$$
• $$f'_y\left( x,y \right)=2x^2y$$
• Alternative notation for $$f'_x\left( x,y \right)$$ :

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

• Alternative notation for $$f'_y\left( x,y \right)$$ :

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

• $$z=f\left( x,y \right)$$ 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

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

• Second partial derivatives:
• 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).