Q.Find $\frac{dy}{dx}$, if $x = a(\theta + \sin\theta)$, $y = a(1 - \cos\theta)$.

Concept: Parametric Differentiation — when both $x$ and $y$ are given in terms of a parameter $\theta$, we use $\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}$.

First, differentiate each with respect to $\theta$:

Now apply the formula:

$\frac{dy}{dx} = \frac{a\sin\theta}{a(1 + \cos\theta)} = \frac{\sin\theta}{1 + \cos\theta}.$ …

When a curve is given in parametric form — $x$ and $y$ each expressed in terms of a third variable (here $\theta$) — we cannot directly write $y$ as a function of $x$. Instead, we use the chain rule in a clever way:

provided $dx/d\theta \neq 0$. This works because both $x$ and $y$ are functions of $\theta$, so the derivative of $y$ with respect to $x$ is the ratio of their individual rates of change with respect to $\theta$.

Let's apply this step by step.

1. Differentiate $x$ with respect to $\theta$ $x = a(\theta + \sin\theta)$ The derivative of $\theta$ is $1$, and the derivative of $\sin\theta$ is $\cos\theta$. So:

2. Differentiate $y$ with respect to $\theta$ $y = a(1 - \cos\theta)$ The derivative of $1$ is $0$, and the derivative of $-\cos\theta$ is $\sin\theta$ (since $\frac{d}{d\theta}(-\cos\theta) = \sin\theta$). So:

3. Form the ratio

4. Simplify using a trigonometric identity Recall the half-angle identities: …

Method: Differentiating Parametric Equations and Simplifying with Half-Angle Identities

This method applies whenever $x$ and $y$ are given parametrically using trig expressions such as $\theta+\sin\theta$ or $1-\cos\theta$, where the raw ratio $\dfrac{dy}{dx}$ needs a trigonometric identity to reach a clean closed form.

Steps

Step 1: Apply the parametric-derivative formula

Step 2: Differentiate each equation with respect to $\theta$

Differentiate $x(\theta)$ and $y(\theta)$ term by term. Terms like $\theta$ contribute $1$; trig terms contribute their standard derivatives ($\sin\theta\to\cos\theta$, $-\cos\theta\to\sin\theta$, etc.).

Step 3: Recognize and apply a half-angle identity …

Common Mistakes

Mistake 1: Leaving the answer as an unsimplified trig ratio

Why it's wrong: $\dfrac{\sin\theta}{1+\cos\theta}$ is mathematically correct but not the expected final form — NCERT-style answers reduce this to $\tan(\theta/2)$ using half-angle identities. Correct approach: always check whether the ratio matches a known half-angle pattern before declaring the derivative finished.

Mistake 2: Misapplying the half-angle identity …