AcademySeveral-Variable Calculus
Academy
Partial Derivatives
Level 1 - Calculus topic page in Several-Variable Calculus.
Principle
A partial derivative measures the rate of change with respect to one input while the other inputs are held constant.
Notation
\(f_x\)
partial derivative with respect to x
\(f_y\)
partial derivative with respect to y
\(\frac{\partial f}{\partial x}\)
same as f_x
The Core Method
To compute \(f_x\), treat \(y\) as a constant and differentiate with respect to \(x\). To compute \(f_y\), treat \(x\) as a constant.
Partial derivative
\[f_x(x,y)=\frac{\partial f}{\partial x}\]
The notation changes, but the differentiation rules are the same.
Worked Cases
Question
For \(f(x,y)=x^2y+3y^2\), find \(f_x\) and \(f_y\).
Answer
For \(f_x\), treat \(y\) as constant: \(f_x=2xy\). For \(f_y\), treat \(x\) as constant: the derivative of \(x^2y\) is \(x^2\), and the derivative of \(3y^2\) is \(6y\). Thus \(f_y=x^2+6y\).
Examples
Question
If \(f_x=2xy\), find \(f_x(3,-1)\).
Answer
Substitute \(x=3\) and \(y=-1\): \(f_x(3,-1)=2(3)(-1)=-6\).