Q.Find $\frac{dy}{dx}$, if $x = a\cos\theta$, $y = a\sin\theta$.

The key idea is that when both $x$ and $y$ are given in terms of a parameter $\theta$, we use the chain rule:

$\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}$.

Step 1: Differentiate $x = a\cos\theta$ with respect to $\theta$:

$\frac{dx}{d\theta} = -a\sin\theta$.

Step 2: Differentiate $y = a\sin\theta$ with respect to $\theta$:

$\frac{dy}{d\theta} = a\cos\theta$.

Step 3: Divide:

$\frac{dy}{dx} = \frac{a\cos\theta}{-a\sin\theta} = -\cot\theta$.

When a curve is given in parametric form — where both $x$ and $y$ are expressed in terms of a third variable (here $\theta$) — you cannot directly differentiate $y$ with respect to $x$ as a single function. Instead, you use the chain rule in reverse:

provided $dx/d\theta \neq 0$. This works because $dy/d\theta = (dy/dx) \cdot (dx/d\theta)$, so solving for $dy/dx$ gives the ratio.

Let’s apply this to the given equations.

1. Differentiate $x$ with respect to $\theta$ $x = a\cos\theta$ The derivative of $\cos\theta$ is $-\sin\theta$, so

2. Differentiate $y$ with respect to $\theta$ $y = a\sin\theta$ The derivative of $\sin\theta$ is $\cos\theta$, so

3. Form the ratio

The $a$ cancels (provided $a \neq 0$, which is true for a non-degenerate circle), leaving

Method: Differentiating Parametric Equations for dy/dx

This method applies whenever both $x$ and $y$ are given as separate functions of a third variable (a parameter such as $\theta$ or $t$), and you need the slope $\frac{dy}{dx}$.

Steps

Step 1: Recognize the parametric form and write down the governing formula

Since $y$ is not written directly as a function of $x$, you cannot differentiate $y$ with respect to $x$ term by term. Instead, use

This follows from the chain rule: $\frac{dy}{d\theta}=\frac{dy}{dx}\cdot\frac{dx}{d\theta}$, so dividing both sides by $\frac{dx}{d\theta}$ isolates $\frac{dy}{dx}$.

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

Treat $\theta$ (or $t$) as the single independent variable and differentiate each equation using the ordinary rules (power, trig, exponential). Do this for $x$ first, then $y$ — keep the two derivatives separate; do not combine them yet.

Step 3: Form the ratio and simplify

Divide the $y$-derivative by the $x$-derivative. Cancel any common constant factors, and if the ratio matches a known trig identity (e.g. $\cot\theta$, $\tan(\theta/2)$), rewrite it in that simplest form — examiners expect the simplified answer, not the raw ratio.

Common Mistakes

Mistake 1: Sign error differentiating $\cos\theta$

Why it's wrong: $\frac{d}{d\theta}(\cos\theta) = -\sin\theta$, and it is easy to drop that negative sign when several terms are in play. Correct approach: always write the derivative of $\cos\theta$ as $-\sin\theta$ explicitly before simplifying, and double-check the sign survives into the final ratio.

Mistake 2: Inverting the ratio

Why it's wrong: the formula is $\frac{dy}{dx}=\dfrac{dy/d\theta}{dx/d\theta}$ — the $y$-derivative on top, $x$-derivative on the bottom. Under time pressure students sometimes flip this. Correct approach: write both derivatives out fully labelled ($dx/d\theta = \ldots$, $dy/d\theta=\ldots$) before dividing, so the labelling itself prevents the inversion.