Q.Find $\frac{dy}{dx}$ in the following: $x = \sin t, y = \cos 2t$

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

First, differentiate each with respect to $t$:

Now apply the chain rule:

Use the identity $\sin 2t = 2\sin t \cos t$ to simplify:

Why parametric differentiation works

When $x$ and $y$ are both given in terms of a third variable $t$, we cannot directly write $y$ as a function of $x$ — and we don't need to. The chain rule gives us a clean way:

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$, provided $dx/dt \neq 0$.

Think of it this way: a small change in $t$ causes a small change in both $x$ and $y$. The ratio of those changes (as $dt \to 0$) is exactly the derivative we want.

Step-by-step solution

1. Differentiate $x$ with respect to $t$

$x = \sin t$

$\frac{dx}{dt} = \cos t$

2. Differentiate $y$ with respect to $t$

$y = \cos 2t$

Using the chain rule: $\frac{dy}{dt} = -\sin(2t) \cdot 2 = -2 \sin 2t$

3. Apply the parametric derivative formula

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-2 \sin 2t}{\cos t}$

4. Simplify using the identity $\sin 2t = 2 \sin t \cos t$

$\frac{dy}{dx} = \frac{-2 (2 \sin t \cos t)}{\cos t} = \frac{-4 \sin t \cos t}{\cos t}$

5. Cancel $\cos t$ (provided $\cos t \neq 0$, i.e., $t \neq \frac{\pi}{2} + n\pi$)

$\frac{dy}{dx} = -4 \sin t$

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

Parametric answers built from $\sin,\cos$ of the parameter very often simplify with a double-angle or half-angle identity ($\sin 2\theta = 2\sin\theta\cos\theta$, $1-\cos\theta = 2\sin^2\frac{\theta}{2}$, $1+\cos\theta = 2\cos^2\frac{\theta}{2}$, etc.) — always look for one before leaving the answer as a raw ratio.

Applying to this problem: with $x=\sin t,\ y=\cos 2t$, the inner derivative of $\cos 2t$ carries a chain-rule factor of $2$ (giving $-2\sin 2t$), and the double-angle identity $\sin 2t = 2\sin t\cos t$ then cancels the $\cos t$ in the denominator cleanly, leaving a one-variable answer in $t$ alone.

Common Mistakes

Mistake 1: Dropping the chain-rule factor of $2$ on $\cos 2t$.

Why it's wrong: $\frac{d}{dt}\cos 2t = -2\sin 2t$, not $-\sin 2t$ — the inner function $2t$ has derivative $2$, which must be multiplied in. Correct approach: always write the chain rule explicitly for a multiple angle before simplifying.

Mistake 2: Cancelling $\cos t$ without excluding where it vanishes.

Why it's wrong: the simplification $\frac{-2\sin 2t}{\cos t}=-4\sin t$ is only valid for $\cos t \neq 0$, i.e. $t \neq \frac{\pi}{2}+n\pi$ — those points are exactly where $dx/dt=0$. Correct approach: state the domain restriction alongside the simplified answer.