Chain Rule for Multivariables Tool

Math Calculus • Multivariable Calculus

Written by STEM Calculators Team Published December 30, 2025 Updated February 24, 2026

8. Chain Rule For Multivariables Tool

Computes total derivatives using multivariable chain rule (single-parameter or two-level dependency tree). Includes a graph of \(z(t)\) (and optional \(dz/dt\)).

Dependency mode:

Outer function

z = f(…): Inner vars (Mode A):

Input supports pi, e, sqrt(2), trig (sin, cos), etc. Use * for multiplication. Drag graph to pan; mouse wheel to zoom; double-click resets.

Inner functions (Mode A)

x(t): y(t):

Chain rule: \(\dfrac{dz}{dt}=f_x\,\dfrac{dx}{dt}+f_y\,\dfrac{dy}{dt}\) (and \(+f_w\,dw/dt\) if used).

Dependency tree (Mode B)

z = f(x,y): x = x(u,v): y = y(u,v): u(t): v(t):

Chain rule (two-level): \[ \frac{dz}{dt}=f_x\Big(x_u u'+x_v v'\Big)+f_y\Big(y_u u'+y_v v'\Big). \]

Evaluation and graph range

t₀ (evaluate at): t min: t max: Graph extra: Samples:

Graph settings (optional)

Show grid: Y padding: Engine preference: Show tree:

Ready

Enter your dependency and click “Calculate”.

Graph

z(t) with axes + tick values

x-axis is \(t\). y-axis is the plotted quantity (z, dz/dt, or both). The marker indicates \(t_0\).

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Theory

Total derivative (single parameter)

If \(z=f(x,y)\) and \(x=x(t)\), \(y=y(t)\), then \(z\) becomes a function of \(t\): \[ z(t)=f(x(t),y(t)). \] The chain rule gives the total derivative: \[ \frac{dz}{dt}=f_x(x(t),y(t))\,\frac{dx}{dt}+f_y(x(t),y(t))\,\frac{dy}{dt}. \]

If \(z=f(x,y,w)\) with \(w=w(t)\), then \[ \frac{dz}{dt}=f_x\,x'(t)+f_y\,y'(t)+f_w\,w'(t). \]

Two-level dependency (u,v intermediates)

Suppose \[ z=f(x,y),\quad x=x(u,v),\quad y=y(u,v),\quad u=u(t),\quad v=v(t). \] First compute \[ \frac{dx}{dt}=x_u\,u'(t)+x_v\,v'(t),\qquad \frac{dy}{dt}=y_u\,u'(t)+y_v\,v'(t), \] then substitute into \[ \frac{dz}{dt}=f_x\,\frac{dx}{dt}+f_y\,\frac{dy}{dt}. \] Combining yields the common “two-level” chain rule: \[ \frac{dz}{dt}=f_x\big(x_u u'+x_v v'\big)+f_y\big(y_u u'+y_v v'\big). \]

Worked example

Let \(z=x^2y\), with \(x=t^2\) and \(y=\sin t\). Then \(f_x=2xy\) and \(f_y=x^2\). Also \(x'=2t\) and \(y'=\cos t\). So \[ \frac{dz}{dt}=f_x x' + f_y y' =(2xy)(2t)+(x^2)(\cos t). \] Substitute \(x=t^2\), \(y=\sin t\): \[ \frac{dz}{dt}=2t^2\sin t + t^4\cos t. \]

Graph interpretation

The calculator plots \(z(t)\) (and optionally \(dz/dt\)) over your chosen \(t\)-interval. The point \(t_0\) is marked on the primary plotted curve.

Frequently Asked Questions

What does dz/dt mean in the multivariable chain rule?

dz/dt is the total derivative of z with respect to t when z depends on intermediate variables like x(t) and y(t). It combines partial derivatives of f with the derivatives of the inner functions.

How is the total derivative different from a partial derivative?

A partial derivative like f_x treats other variables as independent, while the total derivative accounts for how each variable changes with t through the given dependencies. Total derivatives apply when variables are linked by functions of t.

How does the two-level dependency chain rule work?

In Mode B, x and y depend on u and v, and u and v depend on t, so dx/dt and dy/dt are computed first using x_u, x_v, y_u, and y_v, then substituted into dz/dt = f_x dx/dt + f_y dy/dt.

What is the purpose of t0 in this calculator?

t0 sets the point where the tool evaluates the computed expressions, such as dz/dt at t = t0. The graph still shows behavior across the full t interval you choose.

Why would I plot dz/dt as well as z(t)?

Plotting dz/dt shows how fast z changes with t across the interval, while z(t) shows the accumulated function values. Viewing both helps connect the slope behavior to the function’s shape.