Let’s solve the given problem step by step.

Function:

The function provided is: $f(x) = 5x^2 + x + 7$

1. Step 1: Evaluate $f(x+h)$

We substitute $x+h$ into $f(x)$: $f(x+h) = 5(x+h)^2 + (x+h) + 7$

Now expand $(x+h)^2$: $(x+h)^2 = x^2 + 2xh + h^2$

Substitute this back into $f(x+h)$: $f(x+h) = 5(x^2 + 2xh + h^2) + (x + h) + 7$

Distribute the $5$ across $(x^2 + 2xh + h^2)$: $f(x+h) = 5x^2 + 10xh + 5h^2 + x + h + 7$

So, we have: $f(x+h) = 5x^2 + 10xh + 5h^2 + x + h + 7$

2. Step 2: Evaluate $f(x+h) - f(x)$

Now subtract $f(x) = 5x^2 + x + 7$ from $f(x+h)$: $f(x+h) - f(x) = \left(5x^2 + 10xh + 5h^2 + x + h + 7\right) - \left(5x^2 + x + 7\right)$

Simplify by canceling common terms: $f(x+h) - f(x) = 10xh + 5h^2 + h$

3. Step 3: Divide by $h$

Now divide $f(x+h) - f(x)$ by $h$: $\frac{f(x+h) - f(x)}{h} = \frac{10xh + 5h^2 + h}{h}$

Simplify each term: $\frac{f(x+h) - f(x)}{h} = 10x + 5h + 1$

Final Answer:

$\frac{f(x+h) - f(x)}{h} = 10x + 5h + 1$

Let me know if you need further clarifications or explanations!

Related Questions:

1. What is the meaning of the difference quotient, and how is it connected to the derivative?
2. How would this process change if the function $f(x)$ were cubic instead of quadratic?
3. Can you simplify the expression $\frac{f(x+h)-f(x)}{h}$ further if $h \to 0$?
4. What is the derivative of $f(x) = 5x^2 + x + 7$ using differentiation rules?
5. Why do we cancel terms when evaluating $f(x+h) - f(x)$?

Tip:

Always distribute and simplify systematically to avoid errors in algebraic expansion.