Find the derivative of $f(x)=\frac{x+1}{x}$

Step 1: Rewrite the function

First, we can simplify the given function by splitting the fraction into two terms. This makes it easier to apply the power rule for differentiation. We rewrite $\frac{1}{x}$ as $x^{-1}$.

Step 2: Apply the sum rule of differentiation

The derivative of a sum of functions is the sum of their derivatives. So, we can differentiate each term separately.

Step 3: Differentiate each term

The derivative of a constant (like 1) is always 0. For the term $x^{-1}$, we use the power rule, which states that $\frac{d}{dx}(x^n) = nx^{n-1}$. Here, $n = -1$, so we get $-1 \cdot x^{-1-1} = -x^{-2}$.

Step 4: Combine the derivatives

Finally, we combine the derivatives of the individual terms to get the derivative of the original function.

Step 5: Rewrite in positive exponent form

It is good practice to express the final answer with positive exponents, so we rewrite $x^{-2}$ as $\frac{1}{x^2}$.

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