Parametric Equations, Curves, Definition, Differentiation, Worked example - Calculus

Parametric differentiation finds the derivative \(\frac{dy}{dx}\) when both \(x\) and \(y\) are functions of a third variable (the parameter, often \(t\)) by using the formula \(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\). To perform it, you differentiate both \(x\) and \(y\) with respect to the parameter (\(t\)) and then divide the resulting derivatives to find the desired derivative \(\frac{dy}{dx}\).

💡Steps for Parametric Differentiation
• Identify the parametric equations: You will have two equations, one for \(x\) and one for \(y\), both in terms of a parameter, say \(t\).
• Differentiate \(x\) with respect to \(t\): Find \(\frac{dx}{dt}\).
Differentiate \(y\) with respect to \(t\): Find \(\frac{dy}{dt}\).
• Apply the formula: Divide the derivative of \(y\) by the derivative of \(x\) to find \(\frac{dy}{dx}\):
\(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\) This works because, by the chain rule, \(\frac{dy}{dt}=\frac{dy}{dx}\cdot \frac{dx}{dt}\), so \(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\).

💡Example
Given the parametric equations: x = t^3 + t and y = t^2 + 1.
• Differentiate \(x\) and \(y\) with respect to \(t\):
\(\frac{dx}{dt}=3t^{2}+1\)
\(\frac{dy}{dt}=2t\)
• Apply the formula: \(\frac{dy}{dx}=\frac{2t}{3t^{2}+1}\)

💡Why use it?
Parametric differentiation is useful for analyzing complex curves in fields like computer graphics, physics, and engineering, as it allows for the description and analysis of intricate curves that might be difficult to represent explicitly or implicitly.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1tolpJ9mIz-hrSc3BJyC0piUzPQ8cKD3b/view?usp=drive_link
• Answers: https://drive.google.com/file/d/1tiM6iM6LTV0bkAhLvC8ij3bjvbloV5Jj/view?usp=drive_link

💡Chapters:
00:00 Parametric equations and differentiation, with examples
03:39 Second derivative, with example

🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials.

🔔Subscribe: https://rumble.com/user/drofeng
_______________________
⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g
_______________________
💥 Follow us on Social Media 💥
🎵TikTok: https://www.tiktok.com/@drofeng?lang=en
𝕏: https://x.com/DrOfEng
🥊: https://youtube.com/@drofeng