Parametric Derivative Calculator — Theory
1. Parametric equations
A parametric curve gives the coordinates of a point as functions of a parameter \(t\):
\[
x=x(t),\qquad y=y(t).
\]
The point on the curve is
\[
\mathbf r(t)=\langle x(t),y(t)\rangle.
\]
As \(t\) changes, the point moves along the curve.
2. First derivative
For a parametric curve, \(y\) is not always written directly as a function of \(x\). Instead, both
depend on \(t\). The chain rule gives
\[
\frac{dy}{dt}
=
\frac{dy}{dx}\frac{dx}{dt}.
\]
If \(\frac{dx}{dt}\ne0\), then
\[
\frac{dy}{dx}
=
\frac{dy/dt}{dx/dt}.
\]
3. Second derivative
The second derivative is not just \(\frac{d^2y/dt^2}{d^2x/dt^2}\). Instead, we differentiate
\(\frac{dy}{dx}\) with respect to \(t\), then divide by \(\frac{dx}{dt}\).
\[
\frac{d^2y}{dx^2}
=
\frac{d}{dt}\left(\frac{dy}{dx}\right)
\Big/
\frac{dx}{dt}.
\]
This formula is one of the most common mistakes in parametric differentiation.
4. Higher-order derivatives
The same idea can be repeated. Define
\[
y_{[1]}=\frac{dy}{dx}.
\]
Then, for \(n\ge2\),
\[
y_{[n]}
=
\frac{d}{dt}\left(y_{[n-1]}\right)
\Big/
\frac{dx}{dt}.
\]
This gives \(\frac{d^3y}{dx^3}\), \(\frac{d^4y}{dx^4}\), and higher derivatives.
5. Tangent line
At \(t=t_0\), the point on the curve is
\[
(x_0,y_0)=\bigl(x(t_0),y(t_0)\bigr).
\]
If \(m=\left.\frac{dy}{dx}\right|_{t=t_0}\), then the tangent line is
\[
y-y_0=m(x-x_0).
\]
If \(dx/dt=0\) and \(dy/dt\ne0\), then the tangent line may be vertical:
\[
x=x_0.
\]
6. Example: unit circle
The unit circle can be parametrized by
\[
x=\cos t,\qquad y=\sin t.
\]
Differentiate each coordinate:
\[
\frac{dx}{dt}=-\sin t,\qquad
\frac{dy}{dt}=\cos t.
\]
Therefore,
\[
\frac{dy}{dx}
=
\frac{\cos t}{-\sin t}
=
-\cot t.
\]
For the second derivative,
\[
\frac{d^2y}{dx^2}
=
\frac{d}{dt}(-\cot t)\Big/(-\sin t)
=
-\csc^3 t.
\]
7. Example: parabola \(x=t,\ y=t^2\)
If
\[
x=t,\qquad y=t^2,
\]
then
\[
\frac{dx}{dt}=1,\qquad
\frac{dy}{dt}=2t.
\]
So
\[
\frac{dy}{dx}=2t.
\]
Since \(x=t\), this is also
\[
\frac{dy}{dx}=2x.
\]
The second derivative is
\[
\frac{d^2y}{dx^2}=2.
\]
8. Example: semicubical parabola
Let
\[
x=t^2,\qquad y=t^3.
\]
Then
\[
\frac{dx}{dt}=2t,\qquad
\frac{dy}{dt}=3t^2.
\]
For \(t\ne0\),
\[
\frac{dy}{dx}
=
\frac{3t^2}{2t}
=
\frac{3t}{2}.
\]
The second derivative is
\[
\frac{d^2y}{dx^2}
=
\frac{d}{dt}\left(\frac{3t}{2}\right)\Big/(2t)
=
\frac{3}{4t}.
\]
At \(t=0\), this curve has a singular behavior, so ordinary derivative formulas must be interpreted carefully.
10. Common mistakes
- Wrong second derivative: do not use \(\frac{d^2y/dt^2}{d^2x/dt^2}\).
- Ignoring \(dx/dt=0\): the slope may be undefined or vertical.
- Confusing tangent vector and slope: \(\mathbf r'(t)\) is a vector, while \(\frac{dy}{dx}\) is a scalar slope.
- Forgetting the parameter value: derivatives depend on the selected \(t_0\).
- Losing orientation: the tangent vector points in the direction of increasing \(t\).
