How do you find and simplify the difference quotient for the function f(x) = x² + 5, where h is not equal to zero?

To find and simplify the difference quotient for the function f(x) = x² + 5, we follow these steps:

1. Recall the definition of the difference quotient:

   The difference quotient is given by the formula:

   Difference Quotient = \\frac{f(x + h) – f(x)}{h}

2. Calculate f(x + h):

   We start by substituting x + h into the function:

   f(x + h) = (x + h)² + 5 = x² + 2xh + h² + 5

3. Calculate f(x + h) – f(x):

   Next, we find f(x + h) – f(x):

   f(x + h) – f(x) = (x² + 2xh + h² + 5) – (x² + 5)

   After simplification, we get:

   f(x + h) – f(x) = 2xh + h²

4. Substitute into the difference quotient formula:

   Now, we can substitute our result into the difference quotient:

   Difference Quotient = \\frac{2xh + h²}{h}

5. Simplify:

   Next, we simplify the expression:

   Difference Quotient = 2x + h

   Note that this simplification is valid for h ≠ 0, as division by zero is undefined.

Therefore, the simplified difference quotient for the function f(x) = x² + 5, where h is not equal to zero, is:

2x + h

This expression gives a good approximation of the function’s slope at the point x when h is small.