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.
