The difference quotient is \( \square . \begin{array}{l}\frac{f(x+h)-f(x)}{h}\end{array} \)

The difference quotient is a fundamental concept in calculus, particularly in the definition of the derivative. It is expressed as: \[ \frac{f(x+h) - f(x)}{h} \] where: - \( f(x) \) is a function, - \( h \) is a small increment in \( x \), - \( x \) is the point at which we are evaluating the function. ### Explanation of the Difference Quotient 1. **Function Evaluation**: The term \( f(x+h) \) represents the value of the function at the point \( x+h \), which is a small step away from \( x \). 2. **Function Value at \( x \)**: The term \( f(x) \) is simply the value of the function at the point \( x \). 3. **Difference**: The expression \( f(x+h) - f(x) \) calculates the change in the function's value as \( x \) changes to \( x+h \). 4. **Average Rate of Change**: Dividing this difference by \( h \) gives the average rate of change of the function over the interval from \( x \) to \( x+h \). 5. **Limit Process**: As \( h \) approaches 0, the difference quotient approaches the derivative of the function at the point \( x \), which is defined as: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] ### Summary The difference quotient is a way to measure how a function changes as its input changes. It is a crucial step in understanding derivatives and the concept of instantaneous rate of change.