Multivariable Chain Rules and Implicit Surfaces

Add contributions from every changing input

If $z=f(x(t),y(t))$ and the relevant maps are differentiable, then

\frac{dz}{dt}=f_x x'+f_y y'.

If $x,y$ depend on variables $u,v$ , apply the same rule separately for each input: $z_u=f_xx_u+f_yy_u$ . Evaluate outer derivatives at the inner point. The Jacobian matrix organizes these relations as $D(f\circ g)=Df(g)Dg$ , with compatible matrix dimensions.

Visual guide

VISUAL GUIDEA tangent plane touches a spherical surface

Sphere slice x² + z² = 5
Tangent z = 2 − (x − 1)/2

This cross-section fixes y = 2 in the sphere x² + y² + z² = 9. At (x, z) = (1, 2), implicit differentiation gives dz/dx = −1/2. The tangent line shown is the corresponding slice of the 3D tangent plane.

Worked example: a path across a surface

Let $z=x^2y$ , with $x=t^2$ and $y=\sin t$ . The chain rule gives

z'=2xy(2t)+x^2\cos t=4t^3\sin t+t^4\cos t.

Substituting first gives $z=t^4\sin t$ , whose ordinary derivative agrees. Agreement between the two routes is a useful check when both are manageable.

For a temperature field $T(x,y,t)$ experienced by a moving sensor, the total rate is $dT/dt=T_xx'+T_yy'+T_t$ . The explicit time term is separate from motion through spatial variation.

Worked example: solve for a surface height

Suppose $F(x,y,z)=x^2+y^2+z^2-9=0$ . Where $F_z=2z\ne0$ , the implicit function theorem allows a local differentiable height $z(x,y)$ , with

z_x=-F_x/F_z=-x/z,

z_y=-F_y/F_z=-y/z.

At $(1,2,2)$ , the slopes are $-1/2$ and $-1$ . The tangent plane is $z-2=-(x-1)/2-(y-2)$ , equivalently $x+2y+2z=9$ .

At equator points where $z=0$ , this height formula fails because the sphere is vertical relative to the $xy$ -plane. The sphere itself is still smooth there: another nonzero partial can provide a different local graph. Failure of one coordinate representation is not necessarily a singular surface.

Practice

Find $d(xy)/dt$ when $x=e^t,y=t^2$ .
Find $z_x$ for $xz+y+z^2=0$ where $x+2z\ne0$ .
What term is missing from $dT/dt=T_xx'+T_yy'$ when the field itself changes with time?

Show worked solutions

$e^tt^2+2te^t$ .
$F_x=z$ and $F_z=x+2z$ , giving $z_x=-z/(x+2z)$ .
The explicit partial derivative $T_t$ .

Further study

MIT OpenCourseWare: Multivariable Calculus provides a full university course with additional lectures and exercises.