Parametric Derivative Calculator

Calculate the first derivative ($\frac{dy}{dx}$) and second derivative ($\frac{d^2y}{dx^2}$) for parametric equations $x(t)$ and $y(t)$, with step-by-step solutions.

First Derivative ($\frac{dy}{dx}$) Second Derivative ($\frac{d^2y}{dx^2}$) Step-by-Step Solution Numerical Evaluation Curve Plotting
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Enter Parametric Equations in terms of $t$

Enter x as a function of t
Enter y as a function of t
Enter a numeric value to compute dy/dx and d²y/dx² at that point
📖 Supported Input Syntax (click to expand)
sin(t) cos(t) tan(t) exp(t) → eᵗ ln(t) → log(t) sqrt(t) t^2, t^3 pi, e abs(t) t - sin(t)

⚠️ Note: Use exp(t) for eᵗ, not e^t. Use ln(t) or log(t) for natural log.

📌 Popular Examples

Computing $\frac{dx}{dt}$, $\frac{dy}{dt}$, and the second derivative...

Parametric Derivative Results

First Derivative $\frac{dy}{dx}$
Second Derivative $\frac{d^2y}{dx^2}$

Step-by-Step Breakdown

📈 Parametric Curve

Plot of the parametric curve $(x(t), y(t))$ for $t \in [-10, 10]$.

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Parametric Derivatives: The Formulas You Need

Parametric equations define coordinates $(x, y)$ in terms of a third variable, typically $t$ (representing time or a parameter). Finding $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ requires the chain rule applied carefully to each component.

The First Derivative $\frac{dy}{dx}$

The slope of a parametric curve at any point is the ratio of the rate of change of $y$ to the rate of change of $x$ — both with respect to the parameter $t$:

First Derivative Formula
$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$$

The Second Derivative $\frac{d^2y}{dx^2}$

The second derivative gives the concavity of the parametric curve. A common mistake is squaring the first derivative — the correct formula differentiates $\frac{dy}{dx}$ with respect to $t$, then divides by $\frac{dx}{dt}$ again:

Second Derivative Formula
$$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\!\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$$

Common Mistakes to Avoid

Worked Example: Unit Circle

For $x(t) = \cos(t)$, $y(t) = \sin(t)$:

Frequently Asked Questions

A parametric derivative is the derivative $\frac{dy}{dx}$ of a curve defined by parametric equations $x(t)$ and $y(t)$. Rather than having $y$ as a direct function of $x$, both are functions of a parameter $t$. The derivative is computed using the chain rule: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$.

Differentiate $x(t)$ with respect to $t$ to get $\frac{dx}{dt}$, and differentiate $y(t)$ with respect to $t$ to get $\frac{dy}{dt}$. Then divide: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$. This is valid as long as $\frac{dx}{dt} \neq 0$.

The second derivative is $\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\!\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$. You differentiate the first derivative $\frac{dy}{dx}$ (which is a function of $t$) with respect to $t$, then divide by $\frac{dx}{dt}$ again.

The derivative $\frac{dy}{dx}$ is undefined when $\frac{dx}{dt} = 0$. This means the parametric curve has a vertical tangent line at that parameter value $t$. Our calculator detects this automatically and displays a clear warning.

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