Multivariable Chain Rule Calculator

The chain rule for functions of several variables: the total derivative dz/dt when x and y depend on t, or ∂z/∂u and ∂z/∂v when they depend on u and v, each checked against differentiating z directly.

x and y depend on	One variable t (total derivative dz/dt)
z = f(x, y)	x^2 y + y^3
x =	cos(t)
y =	sin(t)

x and y depend on

z = f(x, y)

x =

y =

dz/dt

−2cos(t)·sin(t)^2 + (cos(t)^2 + 3sin(t)^2)·cos(t)

By the chain rule, dz/dt = −2cos(t)·sin(t)^2 + (cos(t)^2 + 3sin(t)^2)·cos(t). Writing z directly in t (z = cos(t)^2·sin(t) + sin(t)^3) and differentiating gives the same.

Exact form. Checked numerically.

Method: Chain rule for partial derivatives
Worked out with: AllCalcHub expression engine
Check: Every step compared at several values of x

∂z/∂x: 2x·y
∂z/∂y: x^2 + 3y^2
z in terms of t: cos(t)^2·sin(t) + sin(t)^3

About the Multivariable Chain Rule Calculator

If z = f(x, y) and x, y are functions of t, then dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt): the total derivative. If x and y depend on u and v, ∂z/∂u = z_x x_u + z_y y_u and ∂z/∂v = z_x x_v + z_y y_v.

The same rule is used for changes of variables and Jacobians.

The formula

dz/dt = ∂z/∂x · dx/dt + ∂z/∂y · dy/dt

Worked example

z = x²y + y³, x = cos t, y = sin t

dz/dt = 2xy(−sin t) + (x² + 3y²) cos t
= cos³t + 3 sin²t cos t − 2 sin²t cos t

Questions people ask

What's the difference between d and ∂?

∂ is a partial derivative (other variables held fixed); d is used when z ends up depending on one variable only.

Why use the chain rule if I can substitute?

It is often shorter, and it works when x and y are only known implicitly or as data.