Q.Find $\frac{dy}{dx}$ in the following: $xy = e^{(x-y)}$

Idea: $y$ is tangled inside the exponential, so differentiate both sides with respect to $x$ treating $y=y(x)$, then solve for $\frac{dy}{dx}$.

Left side (product rule): $\dfrac{d}{dx}(xy) = y + x\dfrac{dy}{dx}$.

Right side (chain rule): $\dfrac{d}{dx}e^{x-y} = e^{x-y}\left(1 - \dfrac{dy}{dx}\right)$.

Set them equal:

Collect the $\frac{dy}{dx}$ terms: …

The equation $xy = e^{(x-y)}$ does not give $y$ explicitly in terms of $x$ — it is buried inside an exponential and also multiplied by $x$. So we use implicit differentiation: treat $y$ as a function $y(x)$, differentiate both sides with respect to $x$, and every time a $y$-term is differentiated, multiply by $\frac{dy}{dx}$ (chain rule). Then collect the $\frac{dy}{dx}$ terms and solve.

Differentiate the left side

$xy$ is a product of $x$ and $y(x)$, so by the product rule:

Differentiate the right side

The exponent is $(x-y)$, whose derivative is $1-\frac{dy}{dx}$, so by the chain rule:

Set equal and solve

Move all $\frac{dy}{dx}$ terms to one side: …

Method: Implicit Differentiation With a Simplification Trick Using the Original Equation

Some implicit-differentiation problems let you simplify the final derivative by substituting the original equation itself back into the answer — always check whether this is possible before calling the answer final.

Steps

Step 1: Differentiate both sides as they stand, term by term

Use the product rule for any $xy$-type term and the chain rule for any exponential/composite term. Every $y$ you differentiate picks up a factor of $\dfrac{dy}{dx}$.

Step 2: Watch the sign inside a composite exponent

If the exponent itself is $x-y$ (or similar), its derivative is $1-\dfrac{dy}{dx}$ — the minus sign belongs to the $y$-term, not the whole expression. Getting this backwards flips the sign of your final answer.

Step 3: Collect $\dfrac{dy}{dx}$ terms and solve

Move every $\dfrac{dy}{dx}$-carrying term to one side, factor, divide — the standard implicit-differentiation finish. …

Common Mistakes

Mistake 1: Getting the sign wrong on the exponent's derivative

Why it's wrong: the exponent is $x-y$, so its derivative with respect to $x$ is $1-\dfrac{dy}{dx}$ — writing $1+\dfrac{dy}{dx}$ (adding instead of subtracting the $y$-derivative) is a very common slip, since it's easy to instinctively add the chain-rule term rather than subtract it. Correct approach: differentiate the exponent as its own mini-expression first ($x-y \to 1-\dfrac{dy}{dx}$) before plugging it into $e^{(x-y)}\cdot(\text{that derivative})$.

Mistake 2: Forgetting the product rule on the left side …