Q.Find $\frac{dy}{dx}$, if $x^{2/3} + y^{2/3} = a^{2/3}$.

Concept: Implicit Differentiation — differentiate both sides with respect to $x$, treating $y$ as a function of $x$.

Step 1: Differentiate term by term:

Step 2: Apply the power rule and chain rule:

Step 3: Solve for $\frac{dy}{dx}$: …

The equation $x^{2/3} + y^{2/3} = a^{2/3}$ looks like a curve — it’s actually a special case of an astroid. But we don’t need geometry here; we just need to find the slope of the tangent at any point on this curve.

The key idea: since $y$ is not written explicitly in terms of $x$, we use implicit differentiation. That means we differentiate both sides of the equation with respect to $x$, remembering that whenever we differentiate a term involving $y$, we multiply by $\frac{dy}{dx}$ (by the chain rule). Then we solve algebraically for $\frac{dy}{dx}$.

Let’s go step by step.

1. Differentiate each term Start with the left side:

The right side is a constant ($a$ is a constant), so its derivative is $0$.

2. Apply the power rule For $x^{2/3}$:

For $y^{2/3}$, treat $y$ as a function of $x$:

That extra factor $\frac{dy}{dx}$ is the chain rule — we differentiate the outer function (power) and then multiply by the derivative of the inner function ($y$ with respect to $x$).

3. Set up the equation Putting it all together:

4. Solve for $\frac{dy}{dx}$ Multiply both sides by $3$ to clear the denominator:

Isolate the term with $\frac{dy}{dx}$:

Divide both sides by $2y^{-1/3}$ (which is the same as multiplying by $\frac{y^{1/3}}{2}$): …

Method: Implicit Differentiation

This method applies whenever an equation mixes $x$ and $y$ together (like $x^{2/3}+y^{2/3}=a^{2/3}$) and $y$ cannot easily be isolated as an explicit function of $x$.

Steps

Step 1: Differentiate both sides of the equation with respect to $x$

Treat $y$ as an unknown function $y(x)$ of $x$, not as a separate independent variable. Differentiate every term on both sides.

Step 2: Apply the chain rule to every $y$-term

Whenever you differentiate a term containing $y$, multiply by $\dfrac{dy}{dx}$:

Use the product rule on any mixed term (like $xy$) that contains both variables. …

Common Mistakes

Mistake 1: Forgetting the $\dfrac{dy}{dx}$ factor on a $y$-term

Why it's wrong: writing $\dfrac{d}{dx}(y^{2/3}) = \tfrac{2}{3}y^{-1/3}$ (missing the chain-rule factor) treats $y$ as if it were $x$. Since $y$ is secretly a function of $x$, every differentiated $y$-term must carry a $\dfrac{dy}{dx}$. Correct approach: before differentiating, ask "is this term in $x$ or $y$?" — any $y$-term always picks up $\dfrac{dy}{dx}$.

Mistake 2: Mishandling the negative fractional exponent when solving for $\dfrac{dy}{dx}$ …