Given the function f(x) =3/x, determine the derivative of f at x= -6 using the limit shown below. do not simplify

The derivative of a function \( f(x) \) at a point \( x = a \) can be defined using the limit definition of the derivative as follows:

\[
f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
\]

Given \( f(x) = \frac{3}{x} \), we need to apply this definition at \( x = -6 \). Let's substitute into the formula:

\[
f'(-6) = \lim_{h \to 0} \frac{f(-6 + h) - f(-6)}{h}
\]

First, calculate \( f(-6 + h) \) and \( f(-6) \):
- \( f(-6 + h) = \frac{3}{-6 + h} \)
- \( f(-6) = \frac{3}{-6} = -\frac{1}{2} \)

Now, substitute these into the limit expression:

\[
f'(-6) = \lim_{h \to 0} \frac{\frac{3}{-6 + h} - \left(-\frac{1}{2}\right)}{h}
\]

This is the unsimplified form of the derivative of \( f(x) = \frac{3}{x} \) at \( x = -6 \) using the limit definition.

Would you like me to proceed with simplifying the expression or need further details?

Here are 5 related questions you might find useful:
1. How do you simplify this limit to find the derivative?
2. What is the significance of using the limit definition for derivatives?
3. Can you apply the same method to find the derivative at another point?
4. What happens to the derivative when \( f(x) \) is a constant multiple of \( 1/x \)?
5. How does the derivative of \( f(x) = 3/x \) behave at other values of \( x \)?

**Tip:** When using the limit definition of the derivative, always plug in the specific function and simplify step-by-step for clarity.

Given the function f(x) = 3/x, determine the derivative of f at x= -6 using the limit shown below. Do not simplify.

Learn how to find the derivative of the function f(x) = 3/x at x = -6 using the limit definition of a derivative. This step-by-step process involves key calculus concepts like limits and unsimplified derivatives, suitable for advanced high school or early college students.

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