Q.Find $\frac{dy}{dx}$ in the following: $x = 2at^2, y = at^4$

Concept: Implicit Differentiation (Parametric Form)

Here, both $x$ and $y$ are given in terms of a parameter $t$, so we use the chain rule:

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

Step 1: Differentiate $x$ with respect to $t$:

$\frac{dx}{dt} = 4at$.

Step 2: Differentiate $y$ with respect to $t$:

$\frac{dy}{dt} = 4at^3$.

Step 3: Divide:

$\frac{dy}{dx} = \frac{4at^3}{4at} = t^2$.

Solution

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

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

3. Form the ratio.

Method: Differentiating Functions Given in Parametric Form

When $x$ and $y$ are both given as functions of a third variable (a parameter, usually $t$) rather than one being written directly in terms of the other, use the parametric chain-rule identity instead of trying to eliminate the parameter first.

Steps

Step 1: Recognise the parametric setup

If you're given $x = x(t)$ and $y = y(t)$ separately, $\dfrac{dy}{dx}$ is not found by differentiating $y$ "with respect to $x$" directly — there's no explicit $y(x)$ to differentiate.

Step 2: Differentiate each equation with respect to the parameter

Find $\dfrac{dx}{dt}$ and $\dfrac{dy}{dt}$ separately, using ordinary differentiation rules on each.

Step 3: Divide, don't invert

This comes from the chain rule: $\dfrac{dy}{dt} = \dfrac{dy}{dx}\cdot\dfrac{dx}{dt}$, rearranged. Always put $dy/dt$ on top — inverting the ratio is the single most common error on parametric problems.

Step 4: Simplify the ratio

Cancel common factors between $dy/dt$ and $dx/dt$ (constants, powers of $t$) to get the derivative in its simplest form, typically still expressed in terms of the parameter $t$ rather than $x$ or $y$ directly — that's the expected final form unless the question asks you to eliminate $t$.