The derivative of \( \dfrac{x+1}{x} \) is \( \boxed{-\dfrac{1}{x^2}} \). While we can use the \textbf{quotient rule}, splitting the expression into two terms provides a simpler approach.

Method 1: Simplify First (Preferred Method)

Rewrite \( \dfrac{x+1}{x} \) as:
\[\begin{align*}\frac{x+1}{x} = \frac{x}{x} + \frac{1}{x} = 1 + \frac{1}{x}.\end{align*}\]

Now differentiate term-by-term:
\[\begin{align*}\frac{d}{dx} \left( 1 + \frac{1}{x} \right) = 0 – \frac{1}{x^2} = -\frac{1}{x^2}.\end{align*}\]

Method 2: Quotient Rule

Using the quotient rule:
\[\begin{align*}\frac{d}{dx} \left( \frac{x+1}{x} \right) = \frac{x \cdot \frac{d}{dx}(x+1) – (x+1) \cdot \frac{d}{dx}(x)}{x^2}.\end{align*}\]

Simplify step-by-step:
\[\begin{align*}= \frac{x \cdot 1 – (x+1) \cdot 1}{x^2} = \frac{x – (x+1)}{x^2} = \frac{-1}{x^2}.\end{align*}\]

Why Method 1 is Better

By simplifying first, Method 1 avoids the complexity of applying the quotient rule, making it faster and less prone to errors. Use the quotient rule only when simplification isn’t possible.

Similar Example

Find the derivative of \( \dfrac{x^2 + 3x}{x^2} \).

Simplify First:
\[\begin{align*}\frac{x^2 + 3x}{x^2} = 1 + \frac{3}{x}.\end{align*}\]

Differentiate:
\[\begin{align*}\frac{d}{dx} \left( 1 + \frac{3}{x} \right) = 0 – \frac{3}{x^2} = -\frac{3}{x^2}.\end{align*}\]

Using Quotient Rule:
\[\begin{align*}\frac{d}{dx} \left( \frac{x^2 + 3x}{x^2} \right) = \frac{x^2 \cdot (2x+3) – (x^2 + 3x) \cdot 2x}{(x^2)^2}.\end{align*}\]

Simplify:
\[\begin{align*}= -\frac{3}{x^2}.\end{align*}\]

Conclusion

Simplifying first saves time and effort. Use this approach whenever possible.