We are tasked with finding and simplifying the difference quotient for the function $f(x) = 7x^2$. The difference quotient is given by:

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

Step-by-step solution:

1. Substitute $f(x) = 7x^2$ into the formula:

   $f(x+h) = 7(x+h)^2$

2. Expand $f(x+h)$:

   $f(x+h) = 7(x^2 + 2xh + h^2) = 7x^2 + 14xh + 7h^2$

3. Now compute $f(x+h) - f(x)$:

   $f(x+h) - f(x) = (7x^2 + 14xh + 7h^2) - 7x^2 = 14xh + 7h^2$

4. Divide by $h$:

   $\frac{f(x+h) - f(x)}{h} = \frac{14xh + 7h^2}{h} = 14x + 7h$

Thus, the simplified difference quotient is:

$\frac{f(x+h) - f(x)}{h} = 14x + 7h$

This expression is simplified, but as $h \to 0$, it will approach $14x$, which is the derivative of the function $f(x)$.

Would you like further details on any step?

Related questions:

1. What is the derivative of $f(x) = 7x^2$ using limits?
2. How is the difference quotient related to the concept of a derivative?
3. How does the difference quotient change for different powers of $x$ in functions?
4. What happens to the difference quotient when $h \to 0$?
5. Can the difference quotient be used to approximate instantaneous rates of change?

Tip:

The difference quotient is a foundational concept in calculus, as it directly leads to the formal definition of a derivative.