Partial Derivative

Partial derivatives are derivatives of multivariate functions (usually bivariate functions) where differentiation is done with respect to a single variable, and the rest are treated as constants. Like the normal derivative, they are defined using first principles.

Definition §

Partial Derivative (Bivariate) ^definition

Let $f(x,y)$ be a bivariate function. The (first-order) partial derivatives of $f$ is given by:

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

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

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

Second-Order §

Let $f(x,y)$ be a bivariate function. The second order partial derivatives of f with respect to x and y are defined by:

$$f_{xx}=\frac{\partial }{\partial x}(\frac{\partial f}{\partial x})=\frac{{\partial }^{2}f}{\partial x^2}$$

$$f_{yy}=\frac{\partial }{\partial y}(\frac{\partial f}{\partial y})=\frac{{\partial }^{2}f}{\partial y^2}$$

Mixed Partial Derivatives §

For second-order and higher partial derivatives, we can first differentiate with respect to (w.r.t) one variable, then differentiate again, but this time w.r.t another, different variable. This results in a mixed partial derivative.

$$f_{xy}=\frac{\partial }{\partial y}(\frac{\partial f}{\partial x})=\frac{{\partial }^{2}f}{\partial y\partial x}$$

$$f_{yx}=\frac{\partial }{\partial x}(\frac{\partial f}{\partial y})=\frac{{\partial }^{2}f}{\partial x\partial y}$$

Theorems §

Examples §

1: Partial derivatives using first principles

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

Solution

$$\begin{array}{rl}{f}_y=\frac{\partial f}{\partial y}& =\underset{h\to 0}{\lim }\frac{f(x,y+h)-f(x,y)}{h}\\ & =\underset{h\to \infty }{\lim }\frac{x(y+h{)}^2-x{y}^2}{h}\\ & =\underset{h\to 0}{\lim }\frac{x{y}^2+2xyh+x{h}^2-x{y}^2}{h}\\ & =\underset{h\to 0}{\lim }\frac{\cancel{h}(2xy+xh)}{\cancel{h}}=\underset{h\to 0}{\lim }(2xy+xh)\\ & =2xy\end{array}\text{\hspace{1em}(Definition)}$$