What is the difference quotient of f if f(x)=x^2 +5x -1?

The difference quotient of a function \( f(x) \) is given by:

\[
\frac{f(x+h) - f(x)}{h}
\]

where \( h \) is a small increment. Given the function \( f(x) = x^2 + 5x - 1 \), we need to find \( f(x+h) \) and then substitute it into the difference quotient formula.

1. **Calculate \( f(x+h) \):**

\[
f(x+h) = (x+h)^2 + 5(x+h) - 1
\]

Expand \( (x+h)^2 \) and \( 5(x+h) \):

\[
f(x+h) = x^2 + 2xh + h^2 + 5x + 5h - 1
\]

2. **Find the difference \( f(x+h) - f(x) \):**

\[
f(x+h) - f(x) = \left( x^2 + 2xh + h^2 + 5x + 5h - 1 \right) - \left( x^2 + 5x - 1 \right)
\]

Simplify by canceling out \( x^2 \), \( 5x \), and \( -1 \):

\[
f(x+h) - f(x) = 2xh + h^2 + 5h
\]

3. **Form the difference quotient:**

\[
\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 + 5h}{h}
\]

Simplify by dividing each term by \( h \):

\[
\frac{f(x+h) - f(x)}{h} = 2x + h + 5
\]

So, the difference quotient for \( f(x) = x^2 + 5x - 1 \) is:

\[
\frac{f(x+h) - f(x)}{h} = 2x + h + 5
\]

Would you like further details on this process or have any questions?

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Here are 5 related questions for further exploration:

1. How does the difference quotient relate to the derivative of a function?
2. What happens to the difference quotient as \( h \) approaches 0?
3. How would you find the derivative of \( f(x) = x^2 + 5x - 1 \) using limits?
4. Can you calculate the difference quotient for a different function, such as \( f(x) = \sin(x) \)?
5. What is the significance of the difference quotient in calculus?

**Tip:** The difference quotient is the foundation for understanding derivatives, which measure the rate of change of a function.

Explore how to calculate the difference quotient for the function f(x) = x^2 + 5x - 1 step-by-step. Understand the relationship between the difference quotient and derivatives, essential for calculus. Suitable for advanced high school students.

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