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Multivariable Chain Rule Calculator

Compute the total derivative of z=f(x(t),y(t)) by the chain rule.

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dz/dt = z_x · dx/dt + z_y · dy/dt
e.g. x^2*y
e.g. t
e.g. t^2

📖 Tutorial | Multivariable Chain Rule

1. Theorem
If z=f(x,y) is differentiable and x=x(t), y=y(t), then dz/dt = z_x·dx/dt + z_y·dy/dt.
2. Symbols
z_x,z_ypartials w.r.t. intermediates
dx/dt, dy/dtspeeds of intermediates
dz/dttotal derivative
3. How it works
  • Find x(t₀), y(t₀);
  • Numerically find z_x, z_y;
  • Numerically find x', y';
  • Multiply and sum.
4. Steps
  1. Enter z, x(t), y(t);
  2. Enter t₀;
  3. Click Calculate.
5. Example
Example: z=x²y, x=t, y=t², t=1.
Solution: z_x=2, z_y=1, x'=1, y'=2; dz/dt=2+2=4.
6. Pitfalls
Do not confuse intermediates with independent variables;
Each path contributes a term;
More intermediates → more terms.

❓ FAQ | Chain Rule

How to remember it?
Sum partial×derivative along each path.
What if z is directly a function of t?
Differentiate directly; results agree.
Why numerical?
No manual expansion needed.
More intermediates?
dz/dt = Σ z_{u_i}·du_i/dt.
Self-check?
Compare with direct substitution.