Calculus How To

Parametric Derivative: Formula, Example

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What is a Parametric Derivative?

If φ = (x(u), y(u)) is the parametric equation for a curve, the parametric derivative of the curve at a point 𝑢0 is the vector:
Φ′ (𝑢0) = [𝑥′ (𝑢0), 𝑦′ (𝑢0)]
parametric derivative

The parametric derivative is a tangent line, with length. The vector’s direction, which points tangent to the curve, depends on the path of the curve. Its magnitude is determined by the tracing speed [1].

How to I Find a Parametric Derivative?

The formula to find a parametric first derivative is:
parametric derivative formula

I used θ here as an example, but you may also differentiate with respect to t or another variable.

Example Question: Find the parametric derivative of the curve defined by x = cos(θ), y = 2sin(θ) when θ = (5π)/6.

Step 1: Calculate the derivative for both functions:

  • x = cos(θ): dx/dθ = -sin (θ)
  • y = 2sin(θ): dy/dθ = 2cos (θ)

Step 2: Plug the derivatives from Step 1 into the formula, along with the given value:
example question

That’s it!

The second derivative formula is a little different from the “usual” second derivative. You can’t just take the derivative of the first derivative. Instead, take the derivative of the first derivative divided by the derivative of the original x term [2]:
second derivative of a parametric function


[1] Kazhdan, M. Parametric Curves. Retrieved July 8, 2021 from:
[2] Math 231E, Lecture 33. Parametric Calculus. Retrieved July 8, 2021 from:

Stephanie Glen. "Parametric Derivative: Formula, Example" From Calculus for the rest of us!

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