Question

If $e^{-y} \cdot y = x$, then $\frac{dy}{dx}$ is

• A. $\frac{y}{1 - y}$
• B. $\frac{1}{xy(1 - y)}$
• C. $\frac{1}{x(1 - y)}$
• D. $\frac{y}{x(1 - y)}$

Hint:

Whenever an equation is already neatly presented as $x = f(y)$, finding $\frac{dx}{dy}$ and flipping it is almost always faster and less prone to algebraic errors than applying implicit differentiation!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given an implicit equation relating variables $x$ and $y$. We need to find the first derivative of $y$ with respect to $x$, denoted as $\frac{dy}{dx}$.

Step 2: Key Formula or Approach:
Instead of differentiating implicitly with respect to $x$ immediately (which can get messy with the product rule and chain rule), we can explicitly express $x$ as a function of $y$: $x = f(y)$.
We can then easily differentiate with respect to $y$ to find $\frac{dx}{dy}$.
Finally, we use the inverse derivative relationship: $\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$.

Step 3: Detailed Explanation:
The given equation is already nicely isolated for $x$:
$$x = y \cdot e^{-y}$$ Differentiate both sides with respect to $y$ using the product rule $(uv)' = u'v + uv'$:
$$\frac{dx}{dy} = \frac{d}{dy}(y) \cdot e^{-y} + y \cdot \frac{d}{dy}(e^{-y})$$ $$\frac{dx}{dy} = (1) \cdot e^{-y} + y \cdot (-e^{-y})$$ Factor out the common term $e^{-y}$:
$$\frac{dx}{dy} = e^{-y}(1 - y)$$ Now, take the reciprocal to find $\frac{dy}{dx}$:
$$\frac{dy}{dx} = \frac{1}{e^{-y}(1 - y)}$$ To match the options, we need to substitute $x$ back into the expression.
From the original equation, we know that $x = y \cdot e^{-y}$, which implies $e^{-y} = \frac{x}{y}$.
Substitute this relation into our derivative:
$$\frac{dy}{dx} = \frac{1}{\left(\frac{x}{y}\right)(1 - y)}$$ Simplify the complex fraction by multiplying the numerator and denominator by $y$:
$$\frac{dy}{dx} = \frac{y}{x(1 - y)}$$

Step 4: Final Answer:
The derivative $\frac{dy}{dx}$ is $\frac{y}{x(1 - y)}$, matching option (D).