# 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

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

• is called the gradient of $$f$$ denoted by $$∇f(x)$$ .