To find the difference quotient of $f(x) = -\frac{1}{x}$ at the point $(-4, \frac{1}{4})$, we use the definition of the difference quotient:

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

Here, we are specifically interested in the point $(-4, \frac{1}{4})$. Thus, we set $x = -4$.

The difference quotient at $x = -4$ is:

$\frac{f(-4 + h) - f(-4)}{h}$

Now we compute $f(-4)$ and $f(-4 + h)$:

$f(-4) = -\frac{1}{-4} = \frac{1}{4}$

$f(-4 + h) = -\frac{1}{-4 + h}$

Substituting these into the difference quotient formula:

$\frac{f(-4 + h) - f(-4)}{h} = \frac{-\frac{1}{-4 + h} - \frac{1}{4}}{h}$

To simplify, we find a common denominator for the fractions in the numerator:

$-\frac{1}{-4 + h} - \frac{1}{4} = -\frac{4 + (-4 + h)}{4(-4 + h)} = -\frac{4 - (4 - h)}{4(-4 + h)} = -\frac{h}{4(-4 + h)}$

Substitute this back into the difference quotient:

$\frac{-\frac{h}{4(-4 + h)}}{h} = -\frac{h}{h \cdot 4(-4 + h)} = -\frac{1}{4(-4 + h)}$

Therefore, the difference quotient for $f(x) = -\frac{1}{x}$ at the point $(-4, \frac{1}{4})$ is:

$\frac{f(-4 + h) - f(-4)}{h} = -\frac{1}{4(-4 + h)}$

Would you like further details or have any other questions?

Here are 5 numbered questions you might ask next:

1. How do you compute the derivative of a function at a point using limits?
2. Can you explain the difference between a difference quotient and a derivative?
3. How do you handle difference quotients for more complex functions?
4. Can you provide an example of finding the difference quotient for a polynomial function?
5. What are some common applications of difference quotients in calculus?

Tip: Practice simplifying complex fractions and finding common denominators to become more comfortable with difference quotient problems.