Q.Find $\frac{dy}{dx}$, if $y = 12 (1 - \cos t)$, $x = 10 (t - \sin t)$, $-\frac{\pi}{2} < t < \frac{\pi}{2}$

The key idea is Implicit Differentiation (or parametric differentiation — we find $\frac{dy}{dx}$ by dividing the derivatives with respect to the parameter $t$).

Step 1: Differentiate $x$ and $y$ with respect to $t$.

Step 2: Use the parametric formula $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$.

Step 3: Simplify using the identity $\sin t = 2 \sin\frac{t}{2} \cos\frac{t}{2}$ and $1 - \cos t = 2 \sin^2\frac{t}{2}$. …

The core idea here is parametric differentiation. When $y$ and $x$ are both expressed in terms of a third variable (here $t$), we cannot directly write $y$ as a function of $x$ in a simple way. Instead, we use the chain rule:

This works because the derivative $\frac{dy}{dx}$ is the rate of change of $y$ with respect to $x$, and we can find both rates of change with respect to $t$ separately, then divide them. The condition $-\frac{\pi}{2} < t < \frac{\pi}{2}$ ensures that $dx/dt$ is not zero (we'll check that), so the division is valid.

Let's work through it step by step.

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

   $y = 12 (1 - \cos t)$.

   The derivative of a constant (12) times a function is 12 times the derivative of the function.

   $\frac{d}{dt}(1 - \cos t) = 0 - (-\sin t) = \sin t$.

   So, $\frac{dy}{dt} = 12 \sin t$.

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

   $x = 10 (t - \sin t)$.

   $\frac{d}{dt}(t - \sin t) = 1 - \cos t$.

   So, $\frac{dx}{dt} = 10 (1 - \cos t)$.

3. Apply the parametric formula.

4. Simplify the fraction. Both numerator and denominator have a common factor of 2: $\frac{12 \sin t}{10 (1 - \cos t)} = \frac{6 \sin t}{5 (1 - \cos t)}$ …

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 …

Common Mistakes

Mistake 1: Sign slip differentiating $-\cos t$ inside $1-\cos t$.

Why it's wrong: $\frac{d}{dt}(1-\cos t) = 0-(-\sin t)=\sin t$ — losing track of the double negative can leave a wrong sign on $dx/dt$. Correct approach: differentiate the constant $1$ and the $-\cos t$ term separately and combine signs carefully.

Mistake 2: Not noting that $t=0$ makes $dx/dt=0$ (a cusp of the cycloid). …