To differentiate $e^{1/x}$, we apply the chain rule, which states that the derivative of a composite function $f(g(x))$ is $f'(g(x)) \cdot g'(x)$.

Steps:

1. Let $u = \frac{1}{x}$. This makes the function $e^{1/x}$ equivalent to $e^u$.
2. The derivative of $e^u$ with respect to $u$ is $e^u$.
3. Differentiate $u = \frac{1}{x}$ with respect to $x$:
   $\frac{d}{dx} \left(\frac{1}{x}\right) = -\frac{1}{x^2}.$
4. Combine these results using the chain rule:
   $\frac{d}{dx} \left(e^{1/x}\right) = e^{1/x} \cdot \left(-\frac{1}{x^2}\right).$
5. Simplify:
   $\frac{d}{dx} \left(e^{1/x}\right) = -\frac{e^{1/x}}{x^2}.$

Final Answer:

$\frac{d}{dx} \left(e^{1/x}\right) = -\frac{e^{1/x}}{x^2}.$

Would you like a detailed explanation of the chain rule or additional examples?

Related Questions:

1. How do you apply the chain rule to functions involving logarithms?
2. What is the derivative of $e^{\sin(x)}$?
3. Can you show the step-by-step process for differentiating $e^{x^2}$?
4. How do you compute higher-order derivatives of $e^{1/x}$?
5. What are common mistakes to avoid when applying the chain rule?

Tip:

Always identify the inner function clearly when using the chain rule; this simplifies the differentiation process and avoids errors.