To find the difference quotient of the function $f(x) = \frac{7}{x^2}$, we need to compute:

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

Step 1: Compute $f(x+h)$

Since $f(x) = \frac{7}{x^2}$, replacing $x$ with $x + h$ gives us:

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

Step 2: Set up the difference quotient

Now, substitute $f(x+h)$ and $f(x)$ into the difference quotient formula:

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

Step 3: Simplify the expression

We need to combine the two fractions in the numerator. The least common denominator (LCD) of $(x+h)^2$ and $x^2$ is $x^2(x+h)^2$, so:

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

Step 4: Expand the terms in the numerator

Expand $(x+h)^2 = x^2 + 2xh + h^2$, so the expression becomes:

$\frac{7x^2 - 7(x^2 + 2xh + h^2)}{h \cdot x^2(x+h)^2}$

Simplify the numerator:

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

Now the difference quotient becomes:

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

Step 5: Factor and simplify

Factor out $h$ from the numerator:

$\frac{h(-14x - 7h)}{h \cdot x^2(x+h)^2}$

Cancel $h$ from the numerator and denominator:

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

Final simplified result:

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

This is the simplified form of the difference quotient.

Would you like further explanation or more details on any step?

Here are some related questions:

1. What is the importance of the difference quotient in calculus?
2. How does the difference quotient relate to the derivative?
3. What happens to the difference quotient as $h \to 0$?
4. Can we compute the derivative of $f(x) = \frac{7}{x^2}$ using the power rule?
5. How do the algebraic properties of fractions affect the simplification of difference quotients?

Tip: The difference quotient is the foundation for understanding the derivative of a function, which measures the rate of change.