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Derivatives of Parametric Curves Calculator

Parametric derivative calculator. Enter x(t), y(t) and t to compute dy/dx=(dy/dt)/(dx/dt) and d²y/dx²—for parametric curves.

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dy/dx = (dy/dt)/(dx/dt) d²y/dx² = (d/dt(dy/dx))/(dx/dt)
Use t
Use t
e.g. π/4≈0.785

📖 Tutorial｜Derivatives of Parametric Curves

1. Definition
For x=x(t), y=y(t): dy/dx=(dy/dt)/(dx/dt); and d²y/dx² = (d/dt(dy/dx))/(dx/dt).
2. Symbols
SymbolMeaning
x(t), y(t)Parametric curve
dx/dt, dy/dtDerivatives w.r.t. t
dy/dxTangent slope
d²y/dx²Second derivative
3. How it works
  • We numerically compute dx/dt and dy/dt;
  • Divide for the first derivative;
  • Differentiate dy/dx w.r.t. t again and divide by dx/dt.
4. Steps
  1. Enter x(t), y(t);
  2. Enter the parameter t;
  3. Click Calculate.
5. Example
Example: x=cos t, y=sin t at t=π/4. dy/dx=−cot(π/4)=−1.
6. Pitfalls
The second derivative is NOT (d²y/dt²)/(d²x/dt²);
You must differentiate dy/dx w.r.t. t first;
When dx/dt=0 the first derivative is undefined.

❓ FAQ｜Derivatives of Parametric Curves

How to compute the second parametric derivative?
First D=dy/dx=(dy/dt)/(dx/dt), then d²y/dx²=(dD/dt)/(dx/dt). Don't divide second derivatives directly.
Why not just divide d²y/dt² by d²x/dt²?
The second derivative is w.r.t. x, not t; you must go through D and differentiate again.
What if dx/dt=0?
The tangent is vertical, so dy/dx is undefined.
Tangent slope of a circle?
x=cos t, y=sin t gives dy/dx=−cot t; at t=π/4 it is −1.
Which functions are supported?
Any expression in t—trig, powers, exponentials.