The difference quotation for f(x)= -1/x at the point [-4, 1/4] is ?

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.

Learn how to compute the difference quotient for the function f(x) = -1/x at the point (-4, 1/4). This problem involves calculus concepts such as the difference quotient formula and is suitable for advanced high school students.

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