Q.Find $\frac{dy}{dx}$ if $x - y = \pi$.

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: The derivative of $x$ is $1$, the derivative of $\pi$ (a constant) is $0$, and by the chain rule $\frac{d}{dx}(y) = \frac{dy}{dx}$:

Step 3: Solve for $\frac{dy}{dx}$:

The problem asks for $\frac{dy}{dx}$ given $x - y = \pi$. At first glance, this looks like a simple linear equation — and it is. But the real point here is to see why implicit differentiation works, even when you could solve for $y$ directly.

If you solve for $y$, you get $y = x - \pi$. Differentiating that gives $\frac{dy}{dx} = 1$ immediately. That’s fine. But the problem is likely designed to test implicit differentiation — a technique you’ll need when solving for $y$ is messy or impossible.

Implicit differentiation says: treat $y$ as a function of $x$ (even if you don’t know its explicit form), and differentiate both sides of the equation term-by-term. The chain rule handles $y$: the derivative of $y$ with respect to $x$ is $\frac{dy}{dx}$.

Let’s walk through it.

1. Start with the given equation:

2. Differentiate both sides with respect to $x$. The derivative of $x$ is $1$. The derivative of $y$ (with respect to $x$) is $\frac{dy}{dx}$, by the chain rule. The derivative of the constant $\pi$ is $0$. So:

3. Solve for $\frac{dy}{dx}$:

Method: Differentiating an Equation Implicitly

Use this method whenever $y$ and $x$ are tied together by an equation, rather than $y$ being written as an explicit formula purely in terms of $x$.

Steps

Step 1: Treat $y$ as an unknown function of $x$

Even though $y$ is not written as $y = (\text{expression in } x)$, mentally regard it as $y(x)$. This is what licenses differentiating a $y$-term at all.

Step 2: Differentiate both sides of the equation term by term with respect to $x$

For any term containing $x$ alone, differentiate normally. For any term containing $y$, the chain rule forces an extra factor of $\dfrac{dy}{dx}$:

A pure constant term differentiates to $0$.

Step 3: Collect every term containing $\dfrac{dy}{dx}$ on one side of the equation

Move all other terms (those without $\dfrac{dy}{dx}$) to the opposite side.

Step 4: Factor out $\dfrac{dy}{dx}$ and solve

Divide through by whatever multiplies $\dfrac{dy}{dx}$ to isolate it.

Applying to this type of problem: the method works identically whether the relation between $x$ and $y$ is a simple straight line or a more complicated curve — only the algebra involved in Steps 2–4 changes with the complexity of the equation.

Common Mistakes

Mistake 1: Writing $\frac{d}{dx}(y) = 1$ instead of $\frac{dy}{dx}$

Why it's wrong: This treats $y$ as if it were literally the variable $x$, which is only valid if $y$ and $x$ were the same symbol. Since $y$ is a separate (implicit) function of $x$, its derivative with respect to $x$ is the unknown quantity $\frac{dy}{dx}$, not automatically $1$. Correct approach: always write the derivative of $y$ as $\frac{dy}{dx}$ and solve for it — never assume it equals $1$ just because the derivative of $x$ itself is $1$.

Mistake 2: Differentiating the constant $\pi$ incorrectly

Why it's wrong: $\pi$ is a fixed number, so its derivative is $0$, not $1$ or $\pi$ itself. A slip here changes $1 - \frac{dy}{dx} = 0$ into an equation with a nonzero right-hand side. Correct approach: recognize $\pi$ as a plain constant, exactly like $5$ or $-2$, and differentiate it to $0$.