Partial Differentiation

Partial Differentiation

Most engineering problems involve functions with more than one independent variable (e.g., stress in a beam depending on load, length, and cross-section).

Partial Derivatives

1. First-Order Partial Derivatives:

   • $\frac{\partial z}{\partial x} = f_x(x, y)$
   • $\frac{\partial z}{\partial y} = f_y(x, y)$

2. Second-Order Partial Derivatives:

   • $\frac{\partial^2 z}{\partial x^2} = f_{xx}$ (Differentiate w.r.t $x$ twice)
   • $\frac{\partial^2 z}{\partial y^2} = f_{yy}$ (Differentiate w.r.t $y$ twice)
   • $\frac{\partial^2 z}{\partial x \partial y} = f_{xy}$ (Differentiate w.r.t $y$, then $x$)
   • $\frac{\partial^2 z}{\partial y \partial x} = f_{yx}$ (Differentiate w.r.t $x$, then $y$)

   Clairaut's Theorem states that if continuous, $f_{xy} = f_{yx}$.

Chain Rule for Several Variables

If $z = f(x, y)$ where $x$ and $y$ are functions of a single variable $t$, then the total derivative of $z$ with respect to $t$ is:

$$\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}$$

Total Differentials

The total differential $dz$ approximates the change in $z$ due to small changes in $x$ ($dx$) and $y$ ($dy$).

$$dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy$$

This is fundamental in error analysis for experiments with multiple measured variables.