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The problem asks to find \( f'(1) \) using the definition of the derivative for the function \( f(x) = \frac{4}{x^2} \). The definition of the derivative is given by the following limit:

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

To find \( f'(1) \), we use this definition at \( x = 1 \):

\[
f'(1) = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h}
\]

### Step-by-step solution:

1. **Step 1**: Compute \( f(1+h) \) and \( f(1) \):
   - \( f(1) = \frac{4}{1^2} = 4 \)
   - \( f(1+h) = \frac{4}{(1+h)^2} \)

2. **Step 2**: Substitute these into the derivative definition:
   \[
   f'(1) = \lim_{h \to 0} \frac{\frac{4}{(1+h)^2} - 4}{h}
   \]

3. **Step 3**: Simplify the expression:
   \[
   f'(1) = \lim_{h \to 0} \frac{\frac{4}{(1+h)^2} - \frac{4(1+h)^2}{(1+h)^2}}{h}
   \]
   \[
   = \lim_{h \to 0} \frac{4\left( \frac{1 - (1+h)^2}{(1+h)^2} \right)}{h}
   \]

4. **Step 4**: Expand and simplify the numerator:
   \[
   = \lim_{h \to 0} \frac{4\left(1 - (1 + 2h + h^2)\right)}{h(1 + h)^2}
   \]
   \[
   = \lim_{h \to 0} \frac{4\left(- 2h - h^2 \right)}{h(1 + h)^2}
   \]
   \[
   = \lim_{h \to 0} \frac{-8h - 4h^2}{h(1 + h)^2}
   \]

5. **Step 5**: Cancel the \( h \) in the numerator and denominator:
   \[
   = \lim_{h \to 0} \frac{-8 - 4h}{(1+h)^2}
   \]

6. **Step 6**: Evaluate the limit as \( h \to 0 \):
   \[
   f'(1) = \frac{-8}{1^2} = -8
   \]

Thus, \( f'(1) = -8 \).

Would you like further details on any step? Here are some related questions to deepen your understanding:

1. How does the definition of the derivative relate to the slope of the tangent line?
2. What would happen if you tried to compute \( f'(x) \) for other values of \( x \)?
3. How can we generalize this method to any rational function?
4. What if the function were \( f(x) = \frac{1}{x^2} \) instead of \( \frac{4}{x^2} \)? How would the answer change?
5. How do the properties of limits help simplify complex derivative expressions?

**Tip**: When simplifying expressions involving limits, always check if common terms in the numerator and denominator can be factored and canceled before evaluating the limit.

If f(x) = 4/x^2, find f'(1), using the definition of derivative. f'(1) is the limit as x → of the expression.

Learn how to find the derivative of f(x) = 4/x^2 using the definition of derivative, including step-by-step solutions and a focus on limits and functions. Suitable for students in Grades 11-12 studying calculus.

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