AcademySeveral-Variable Calculus
Academy
Multivariable Chain Rule
Level 1 - Calculus topic page in Several-Variable Calculus.
Principle
The multivariable chain rule differentiates a function whose inputs themselves depend on another variable. It adds the contributions through each input path.
Notation
\(z=f(x,y)\)
dependent variable as a function of x and y
\(x(t),y(t)\)
input variables depending on t
\(\frac{dz}{dt}\)
total derivative with respect to t
The Core Method
If \(z=f(x,y)\), with \(x=x(t)\) and \(y=y(t)\), then
Multivariable chain rule
\[\frac{dz}{dt}=f_x\frac{dx}{dt}+f_y\frac{dy}{dt}\]
Each term is one route by which changing \(t\) changes \(z\).
Worked Cases
Question
Let \(z=x^2+y^2\), \(x=t\), and \(y=t^2\). Find \(dz/dt\).
Answer
Compute partial derivatives: \(f_x=2x\), \(f_y=2y\). Also \(dx/dt=1\), and \(dy/dt=2t\). Thus \(dz/dt=2x(1)+2y(2t)\). Substitute \(x=t\), \(y=t^2\): \(dz/dt=2t+4t^3\).
Examples
Question
Why are there two terms in \(f_xx'(t)+f_yy'(t)\)?
Answer
The output changes when \(x\) changes and when \(y\) changes. The term \(f_xx'(t)\) measures the contribution through \(x\), and \(f_yy'(t)\) measures the contribution through \(y\). The total derivative adds both effects.