Polar Coordinates, Polar Curves, Differentiation - Calculus

To differentiate a polar curve \(r=f(\theta )\), first convert it to parametric equations: \(x=r\cos \theta \) and \(y=r\sin \theta \). Then, find the derivatives \(dx/d\theta \) and \(dy/d\theta \) using the product rule, since \(r\) is a function of \(\theta \). The derivative of the polar curve, \(\frac{dy}{dx}\), is found by dividing \(\frac{dy}{d\theta }\) by \(\frac{dx}{d\theta }\).

💡Step 1: Convert to parametric equations
• The polar equation is given by \(r=f(\theta )\).
• Use the conversion formulas to write \(x\) and \(y\) in terms of \(\theta \):
◦ \(x=r\cos \theta =f(\theta )\cos \theta \)
◦ \(y=r\sin \theta =f(\theta )\sin \theta \)

💡Step 2: Differentiate with respect to \(\theta \)
• Find \(\frac{dy}{d\theta }\) and \(\frac{dx}{d\theta }\) using the product rule, as both \(r\) and the trigonometric functions are changing with respect to \(\theta \).
◦ \(\frac{dx}{d\theta }=\frac{dr}{d\theta }\cos \theta -r\sin \theta \)
◦ \(\frac{dy}{d\theta }=\frac{dr}{d\theta }\sin \theta +r\cos \theta \)

💡Step 3: Find the derivative \(\frac{dy}{dx}\)
• Divide \(\frac{dy}{d\theta }\) by \(\frac{dx}{d\theta }\) to get the slope of the tangent line to the polar curve.
• \(\frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }=\frac{\frac{dr}{d\theta }\sin \theta +r\cos \theta }{\frac{dr}{d\theta }\cos \theta -r\sin \theta }\)

💡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 Polar coordinates, definition
01:35 Polar curves
03:04 Polar differentiation
04:56 Worked example

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