Partial Derivative

Partial derivatives are derivatives of multivariable functions, where one or more variables are treated as constants.

Definition §

Let $f(x,y)$ be the function of two derivatives. The first order partial derivatives of f with respect to the variables x and y can be defined with first principles:

$$f_x=\frac{\partial f}{\partial x}=\underset{h\to 0}{\lim }\frac{f(x+h,y)-f(x,y)}{h}$$

$$f_y=\frac{\partial f}{\partial y}=\underset{h\to 0}{\lim }\frac{f(x,y+h)-f(x,y)}{h}$$

• Note how in the partial derivative in respect to x ($f_x$), treats y as a constant, and vice versa for $f_y$.

Examples §

Using first principles

Let $f(x,y)=x{y}^2$. Find $\frac{\partial f}{\partial y}$ using first principles

$f_y=\frac{\partial f}{\partial y}={\lim }_{h\to 0}\frac{f(x,y+h)-f(x,y)}{h}$ (Definition of partial derivatives)

$f_y={\lim }_{h\to \infty }\frac{x(y+h{)}^2-x{y}^2}{h}$

${\lim }_{h\to 0}\frac{x{y}^2+2xyh+x{h}^2-x{y}^2}{h}$

${\lim }_{h\to 0}\frac{h(2xy+xh)}{h}$

${\lim }_{h\to 0}(2xy+xh)$

$f_y=2xy$