Q.Find $\dfrac{dy}{dx}$ of the function expressed in parametric form: $x = 3\cos\theta - 2\cos^3\theta,\ y = 3\sin\theta - 2\sin^3\theta$.

The key idea is parametric differentiation: when $x$ and $y$ are given in terms of a parameter $\theta$, we use

Step 1 – Differentiate $x$ with respect to $\theta$

Factor:

Step 2 – Differentiate $y$ with respect to $\theta$

Factor:

Step 3 – Form the ratio …

Given $x = 3\cos\theta - 2\cos^3\theta$ and $y = 3\sin\theta - 2\sin^3\theta$, differentiate each with respect to $\theta$.

Differentiate $x$ (chain rule on $\cos^3\theta$ gives $3\cos^2\theta\cdot(-\sin\theta)$):

Differentiate $y$:

$\frac{dy}{d\theta} = 3\cos\theta - 6\sin^2\theta\,\cos\theta = 3\cos\theta\,(1 - 2\sin^2\theta) = 3\cos\theta\,\cos 2\theta.$ …

Method: Parametric Differentiation with Power-of-Trig-Function Terms

Use this when $x(t)$ or $y(t)$ contains a power of sine or cosine (e.g. $\cos^3\theta$) — differentiating that term needs the Chain Rule, and the result usually simplifies dramatically via a double-angle identity.

Steps

Step 1: Differentiate each power term with the Chain Rule

For a term like $\cos^{3}\theta$, treat it as $(\cos\theta)^3$:

Step 2: Combine terms and factor

After differentiating the full expression for $x(\theta)$ (and separately $y(\theta)$), factor out any common trig factor (often $\sin\theta$ or $\cos\theta$) — this sets up the next simplification.

Step 3: Apply a double-angle identity to the bracketed factor …

Common Mistakes

Mistake 1: Missing the inner chain-rule factor when differentiating $\cos^3\theta$

Why it's wrong: writing $\dfrac{d}{d\theta}\cos^3\theta=3\cos^2\theta$ and stopping ignores that $\cos\theta$ itself depends on $\theta$. Correct approach: multiply by the inner derivative, $3\cos^2\theta\cdot(-\sin\theta)$, which is what produces the $6\cos^2\theta\sin\theta$ term.

Mistake 2: Not recognizing the double-angle identity to simplify and cancel

Why it's wrong: leaving the answer as $\dfrac{3\cos\theta(1-2\sin^2\theta)}{3\sin\theta(2\cos^2\theta-1)}$ without spotting that $1-2\sin^2\theta=2\cos^2\theta-1=\cos2\theta$ misses the intended clean cancellation. Correct approach: rewrite both bracketed factors as $\cos2\theta$, which then cancels top and bottom, leaving simply $\cot\theta$. …