The problem asks you to find and simplify the difference quotient for the function $f(x) = 8x^3 + 5x$. The difference quotient is given by:

$\frac{f(x+h) - f(x)}{h}, \quad h \neq 0$

Steps to solve:

1. Substitute $f(x)$ into the difference quotient formula: $f(x) = 8x^3 + 5x$ Now, compute $f(x+h)$ by substituting $x + h$ for $x$ in $f(x)$: $f(x + h) = 8(x+h)^3 + 5(x+h)$

2. Expand $f(x+h)$: Expand $(x+h)^3$ first: $(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$ So, $f(x+h) = 8(x^3 + 3x^2h + 3xh^2 + h^3) + 5(x + h)$ Expanding this gives: $f(x+h) = 8x^3 + 24x^2h + 24xh^2 + 8h^3 + 5x + 5h$

3. Set up the difference quotient: Now compute $f(x+h) - f(x)$: $f(x+h) - f(x) = (8x^3 + 24x^2h + 24xh^2 + 8h^3 + 5x + 5h) - (8x^3 + 5x)$ Simplifying this expression: $f(x+h) - f(x) = 24x^2h + 24xh^2 + 8h^3 + 5h$

4. Divide by $h$: Now, divide the result by $h$: $\frac{f(x+h) - f(x)}{h} = \frac{24x^2h + 24xh^2 + 8h^3 + 5h}{h}$ Factor out $h$ from the numerator: $\frac{f(x+h) - f(x)}{h} = 24x^2 + 24xh + 8h^2 + 5$

Final Answer:

$\frac{f(x+h) - f(x)}{h} = 24x^2 + 24xh + 8h^2 + 5$

This is the simplified form of the difference quotient for the given function $f(x) = 8x^3 + 5x$.

Would you like further details or have any questions?

Related Questions:

1. What happens to the difference quotient as $h$ approaches zero?
2. How does the difference quotient relate to the derivative of a function?
3. How do you calculate the derivative of $f(x) = 8x^3 + 5x$ directly?
4. Can you explain how to expand $(x+h)^3$ step by step?
5. What is the geometric interpretation of the difference quotient?

Tip:

The difference quotient is the foundation for understanding derivatives, as it represents the average rate of change of the function over an interval.