How to evaluate difference quotient?
Understand the Problem
The question is asking how to evaluate the difference quotient, which is a concept in calculus used to find the slope of a secant line between two points on a function. To solve this, we will define the difference quotient and outline the steps to compute it, typically by using the formula (f(x+h)  f(x)) / h as h approaches 0.
Answer
The difference quotient evaluates to the derivative $f'(x)$ as $h$ approaches 0.
Answer for screen readers
The final answer will provide the value of the slope of the secant line as $h$ approaches 0, which is the derivative of the function at point $x$, represented as $f'(x)$.
Steps to Solve

Define the Function and Points Identify the function $f(x)$ for which you want to evaluate the difference quotient and choose a specific point $x$.

Apply the Difference Quotient Formula Use the difference quotient formula, which is given by
$$ \frac{f(x+h)  f(x)}{h} $$
Here, $h$ represents a small change in $x$.

Calculate $f(x+h)$ and $f(x)$ Substitute the point $x$ and the increment $h$ into the function to compute values for $f(x+h)$ and $f(x)$.

Plug into the Formula Insert the values of $f(x+h)$ and $f(x)$ into the difference quotient formula.

Simplify the Expression Simplify the resulting expression from step 4 to eliminate $h$ from the denominator, if possible.

Evaluate the Limit as $h$ Approaches 0 Finally, find the limit of the simplified expression as $h$ approaches 0 to determine the slope of the secant line.
The final answer will provide the value of the slope of the secant line as $h$ approaches 0, which is the derivative of the function at point $x$, represented as $f'(x)$.
More Information
The difference quotient is essential in calculus as it leads to the derivative, representing the instantaneous rate of change of the function at a particular point. Understanding how to evaluate it is fundamental for further studies in calculus.
Tips
 Forgetting to simplify the expression properly, which can lead to not finding the derivative correctly.
 Confusing the difference quotient with the average rate of change; be sure to clarify the distinction.
 Not applying the limit correctly as $h$ approaches 0, which is a crucial step in finding the derivative.