In one line
A partial derivative measures how a function of several variables changes when only one variable changes and all the others are held constant.
Explained simply
Think of the temperature in a room. It depends on where you stand: your position east–west (\(x\)) and north–south (\(y\)). If you walk only towards the east, keeping your north–south position fixed, the rate at which the temperature changes is the partial derivative with respect to \(x\). Walking only north gives the partial derivative with respect to \(y\). Each one tells you about one direction at a time.
Technically, for \(u=f(x,y)\), \(\dfrac{\partial u}{\partial x}=\lim_{h\to0}\dfrac{f(x+h,y)-f(x,y)}{h}\), with \(y\) treated as a constant. Similarly \(\dfrac{\partial u}{\partial y}\) keeps \(x\) constant. Other notations: \(u_x\), \(f_x\), \(\dfrac{\partial f}{\partial x}\). Geometrically, \(u_x\) is the slope of the surface \(u=f(x,y)\) along a line parallel to the \(x\)-axis. For a function of three variables \(u=f(x,y,z)\) there are three first-order partial derivatives \(u_x,u_y,u_z\).
How it works
- Decide which variable you differentiate with respect to.
- Treat every other variable exactly like a fixed number (like 5 or \(a\)).
- Apply ordinary rules: power, product, quotient, chain rule.
- Repeat for each variable to get all first-order partial derivatives.
Key formula
\(\dfrac{\partial u}{\partial x}=\lim_{h\to0}\dfrac{f(x+h,y)-f(x,y)}{h}\), \(\ \dfrac{\partial u}{\partial y}=\lim_{k\to0}\dfrac{f(x,y+k)-f(x,y)}{k}\).
The units of \(u_x\) are (units of \(u\)) per (unit of \(x\)), e.g. °C per metre for temperature.
Quick example
Let \(u=x^2y+\sin(xy)\).
- \(u_x\): treat \(y\) as constant. \(u_x=2xy+y\cos(xy)\).
- \(u_y\): treat \(x\) as constant. \(u_y=x^2+x\cos(xy)\).
At the point \((1,0)\): \(u_x=0+0=0\) and \(u_y=1+1=2\). So at that point, moving in the \(y\) direction changes \(u\) about twice as fast as \(y\) changes, while a small move in the \(x\) direction hardly changes \(u\).
Common mistakes
- Forgetting the chain rule factor: \(\dfrac{\partial}{\partial x}\sin(xy)=y\cos(xy)\), not \(\cos(xy)\).
- Differentiating the "constant" variable as well, e.g. writing \(\dfrac{\partial}{\partial x}(x^2y)=2xy+x^2\).
- Using \(d\) instead of \(\partial\); for several variables the curly \(\partial\) is the correct notation.
How MU asks it
Rarely asked alone; it is the first step of every Module 3 question. Short 3-mark parts like "If \(u=x^y\), find \(u_x\) and \(u_y\)" (answers \(yx^{y-1}\) and \(x^y\log x\)) do appear. Write each derivative on its own line, mention which variable is held constant, and simplify.