Q.Find $\frac{dy}{dx}$ in the following: $x = a (\cos \theta + \theta \sin \theta), y = a (\sin \theta - \theta \cos \theta)$

Concept: Parametric Differentiation — we find $\frac{dy}{dx}$ as $\frac{dy/d\theta}{dx/d\theta}$.

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

Next, differentiate $y$ with respect to $\theta$:

$\frac{dy}{d\theta} = a(\cos\theta - \cos\theta + \theta\sin\theta) = a\theta\sin\theta$ …

We are given $x$ and $y$ as functions of a parameter $\theta$, not directly as $y = f(x)$. This is a classic parametric differentiation problem. The chain rule tells us that if both $x$ and $y$ are differentiable functions of $\theta$, then

provided $dx/d\theta \neq 0$. The intuition is simple: a small change in $\theta$ causes a small change in both $x$ and $y$; the ratio of those changes (as the change shrinks to zero) gives the slope of the curve at that point.

Let’s compute each derivative carefully.

1. Differentiate $x$ with respect to $\theta$. $x = a(\cos \theta + \theta \sin \theta)$. The constant $a$ factors out. Differentiate term by term:
   • Derivative of $\cos \theta$ is $-\sin \theta$.
   • For $\theta \sin \theta$, use the product rule: derivative is $(1)(\sin \theta) + (\theta)(\cos \theta) = \sin \theta + \theta \cos \theta$. So

2. Differentiate $y$ with respect to $\theta$. $y = a(\sin \theta - \theta \cos \theta)$. Again $a$ factors out. Differentiate:
   • Derivative of $\sin \theta$ is $\cos \theta$.
   • For $-\theta \cos \theta$, product rule: derivative is $-(1)(\cos \theta) - (\theta)(-\sin \theta) = -\cos \theta + \theta \sin \theta$. So

3. Form the ratio. $\frac{dy}{dx} = \frac{a \theta \sin \theta}{a \theta \cos \theta} = \frac{\sin \theta}{\cos \theta} = \tan \theta,$ …

Method: Differentiating a Parametric Curve

This method applies whenever a curve is given through a third variable (a parameter — commonly $t$ or $\theta$) instead of $y$ written directly as a function of $x$.

Steps

Step 1: Recognise the parametric form

If you are given $x = f(\text{param})$ and $y = g(\text{param})$ instead of $y = h(x)$, do not try to eliminate the parameter first — it is often messy or impossible. Differentiate each equation separately with respect to the parameter instead.

Step 2: Differentiate $x$ and $y$ with respect to the parameter

Use the ordinary rules (product rule, chain rule, standard derivatives) to find $\dfrac{dx}{d\theta}$ (or $\dfrac{dx}{dt}$) and $\dfrac{dy}{d\theta}$ (or $\dfrac{dy}{dt}$).

Step 3: Divide — the parametric-derivative formula

This is justified by the chain rule: $\dfrac{dy}{d\theta} = \dfrac{dy}{dx}\cdot\dfrac{dx}{d\theta}$, so dividing recovers $\dfrac{dy}{dx}$.

Step 4: Simplify with trigonometric identities where possible …

Common Mistakes

Mistake 1: Forgetting the product rule on $\theta\sin\theta$ or $\theta\cos\theta$, treating $\theta$ as a constant.

Why it's wrong: $\theta$ is the variable of differentiation here, not a fixed constant like $a$ — skipping the product rule loses the entire $\theta\cos\theta$ or $\theta\sin\theta$ contribution. Correct approach: apply $\frac{d}{d\theta}(\theta\cdot f(\theta)) = f(\theta) + \theta f'(\theta)$ explicitly.

Mistake 2: Mis-tracking the sign when differentiating the term $-\theta\cos\theta$. …