Parametric Derivative Tool

Math Calculus • Derivatives

Written by STEM Calculators Team Published December 22, 2025 Updated February 24, 2026

6. Parametric Derivative Tool

Computes \(\dfrac{dy}{dx}\) and \(\dfrac{d^2y}{dx^2}\) for parametric curves \(x(t),y(t)\), and plots the curve with velocity vectors. Drag to pan • wheel/pinch to zoom • animate the parameter \(t\).

Inputs

\(x(t)\)

Use variable t. Supported: + − * / ^, parentheses, constants pi, e, sin cos tan, ln log, exp, sqrt, abs. Implicit multiplication: 2t, 2sin(t), (t+1)(t-1). Trig powers like cos^2(2t) work.

\(y(t)\)

Example above is a cycloid: \(x=t-\sin t,\ y=1-\cos t\).

\(t_{\min}\)

\(t_{\max}\)

Output format

Options Show grid Velocity vectors Many vectors along curve Compute \(\dfrac{d^2y}{dx^2}\)

Evaluate at \(t_0\)

Used for numerical slope + the highlighted point.

\(t_0\) slider

After you compute, the slider maps across \([t_{\min},t_{\max}]\).

Velocity arrow length (px)

Arrow is normalized for visibility.

Animation Animate \(t\)

Runs \(t_0\) from \(t_{\min}\) to \(t_{\max}\) repeatedly.

Quick presets

Click to auto-fill and compute.

Ready

Parametric plot

Drag to pan • wheel/pinch to zoom • Auto fit frames the curve.

x: 0, y: 0, zoom(px/unit): 60

Result

Enter \(x(t)\), \(y(t)\), then click Compute.

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

6. Parametric Derivative Tool — Theory

A parametric curve is defined by giving \(x\) and \(y\) as functions of a parameter \(t\): \[ x=x(t), \qquad y=y(t). \] As \(t\) varies over an interval \([t_{\min},t_{\max}]\), the point \((x(t),y(t))\) traces a curve in the plane.

1) First derivative \(\dfrac{dy}{dx}\) for parametric curves

If \(x(t)\) and \(y(t)\) are differentiable and \(dx/dt \ne 0\), then the slope of the tangent line to the curve is

\[ \boxed{ \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}} }. \]

If \(\dfrac{dx}{dt}=0\) at some \(t\), then \(\dfrac{dy}{dx}\) is undefined there. Geometrically, that often corresponds to a vertical tangent (or a cusp/turning behavior depending on \(dy/dt\)).

2) Second derivative \(\dfrac{d^2y}{dx^2}\)

The second derivative describes how the slope changes as you move along the curve. For parametric curves, differentiate \(\dfrac{dy}{dx}\) with respect to \(t\), then divide by \(dx/dt\):

\[ \boxed{ \frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}} }. \]

Again, if \(dx/dt=0\) then this expression can blow up or become undefined.

3) Velocity vectors (motion interpretation)

If you interpret \((x(t),y(t))\) as the position of a moving particle, then the velocity vector is

\[ \vec v(t)=\left\langle \frac{dx}{dt},\,\frac{dy}{dt} \right\rangle. \]

The velocity vector points in the tangent direction along the curve. Its magnitude \[ |\vec v(t)|=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2} \] is the speed.

4) Worked example: cycloid

Consider the cycloid parameterization: \[ x(t)=t-\sin t,\qquad y(t)=1-\cos t. \]

\[ \frac{dx}{dt}=1-\cos t,\qquad \frac{dy}{dt}=\sin t \] \[ \frac{dy}{dx} = \frac{\sin t}{1-\cos t} \]

5) What the calculator does

Parses \(x(t)\) and \(y(t)\), computes \(dx/dt\) and \(dy/dt\) symbolically.
Forms \(\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}\) and simplifies where possible.
Optionally computes \(\dfrac{d^2y}{dx^2}\) using \(\dfrac{d}{dt}(\dfrac{dy}{dx})/(dx/dt)\).
Draws the parametric curve and velocity vectors; highlights the point at \(t=t_0\).
Allows pan/zoom and animation of \(t\) to visualize motion along the curve.

6) Common issues

Vertical tangents: if \(dx/dt=0\), then \(\dfrac{dy}{dx}\) is undefined (often appears as \(+\infty\), \(-\infty\), or DNE numerically).
Domain restrictions: expressions like \(\sqrt{\cdot}\) and \(\ln(\cdot)\) require valid domains for all sampled \(t\) values.
Discontinuities: if the curve jumps, the plot will show breaks where values are not finite.

Frequently Asked Questions

What is the formula for dy/dx for parametric equations?

For x = x(t) and y = y(t), the slope is dy/dx = (dy/dt)/(dx/dt) as long as dx/dt is not zero. This comes from the chain rule relating changes in y and x through the parameter t.

How do you find the second derivative d2y/dx2 for a parametric curve?

Compute dy/dx first, differentiate it with respect to t, and then divide by dx/dt: d2y/dx2 = (d/dt(dy/dx))/(dx/dt). If dx/dt equals zero, the expression can be undefined.

Why is dy/dx undefined at some values of t?

dy/dx is undefined when dx/dt = 0 because the slope formula divides by dx/dt. Geometrically, this often indicates a vertical tangent, cusp, or a turning behavior depending on dy/dt.

How should I choose tmin and tmax for a parametric plot?

Pick tmin and tmax to cover the interval where the curve segment of interest is traced. If the curve repeats or has singular points, adjust the interval to avoid invalid domains (such as ln of a nonpositive value or sqrt of a negative value).

What do the velocity vectors mean on a parametric curve?

The velocity vector is v(t) = <dx/dt, dy/dt>, which points in the direction of motion along the curve. Its direction matches the tangent direction, and its magnitude represents speed along the parameterized path.

Rate this calculator

Frequently Asked Questions

Related calculators