Q.Derivative of $x^2$ w.r.t. $x^3$ is __________.

Concept: Derivative Evaluation — Here we differentiate one function with respect to another, not with respect to the standard variable $x$. The key is to use the chain rule in the form $\frac{d(x^2)}{d(x^3)} = \frac{d(x^2)/dx}{d(x^3)/dx}$.

Step 1: Differentiate $x^2$ with respect to $x$:

$\frac{d}{dx}(x^2) = 2x$.

Step 2: Differentiate $x^3$ with respect to $x$: …

Concept and Intuition

When we say "derivative of $y$ with respect to $u$", we mean $\frac{dy}{du}$ — the rate at which $y$ changes as $u$ changes. Here, $y = x^2$ and $u = x^3$. The catch is that both are functions of $x$, not directly of each other. So we need a bridge.

The chain rule gives us exactly that bridge:

Think of it this way: if you know how $y$ changes with $x$, and how $u$ changes with $x$, then the ratio of those rates tells you how $y$ changes per unit change in $u$. It’s like converting speeds: if a car travels 60 km per hour and its fuel gauge drops 5 litres per hour, then the fuel consumption is $\frac{5}{60}$ litres per km.

Step-by-Step Solution

1. Identify the functions

   We have $y = x^2$ and $u = x^3$. We want $\frac{dy}{du}$.

2. Differentiate each with respect to $x$

3. Apply the chain-rule formula

4. Simplify Cancel one $x$ (provided $x \neq 0$): …

Method: Differentiating One Function with Respect to Another Function

This method solves "find $\dfrac{du}{dv}$" problems, where you need the rate of change of one expression relative to a second expression — both written in terms of $x$ — rather than the derivative with respect to $x$ itself.

Steps

Step 1: Recognise the target

You are asked for $\dfrac{du}{dv}$ where $u=u(x)$ and $v=v(x)$ are both functions of the same variable $x$, not for $\dfrac{du}{dx}$ or $\dfrac{dv}{dx}$ directly.

Step 2: Differentiate each function separately with respect to $x$

Use the standard differentiation rules (power rule, chain rule, etc.) to find $\dfrac{du}{dx}$ and $\dfrac{dv}{dx}$ independently.

Step 3: Form the ratio

This is the chain rule written in reverse: …

Common Mistakes

Mistake 1: Ignoring "with respect to $x^3$" entirely

A student sees $x^2$ and reflexively answers $2x$, treating the question as an ordinary $\frac{d}{dx}$ problem. Why it's wrong: the question asks for the rate of change of $x^2$ relative to $x^3$, not relative to $x$ — these are different "speeds" being compared. Correct approach: recognize this as $\frac{d(x^2)}{d(x^3)}$, which needs the ratio-of-derivatives (chain rule) technique, not a direct derivative.

Mistake 2: Inverting the ratio

Some students write $\frac{d(x^3)/dx}{d(x^2)/dx}=\frac{3x^2}{2x}$ instead of the correct $\frac{d(x^2)/dx}{d(x^3)/dx}$. Why it's wrong: the quantity being differentiated (the numerator function) must stay on top of the ratio — swapping it answers a completely different question ("derivative of $x^3$ w.r.t. $x^2$"). Correct approach: always put the derivative of the function named FIRST in the problem statement on top. …