If `f(x)=x^(2)-4` then find f'(2).

Answer

To find \( f'(2) \) for the function \( f(x) = x^2 - 4 \), we will use the definition of the derivative:

\[ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \]

In this case, \( a = 2 \).

### Step 1: Calculate \( f(2) \)

First, we need to find \( f(2) \):

\[ f(2) = 2^2 - 4 = 4 - 4 = 0 \]

### Step 2: Substitute into the derivative formula

Now we substitute \( f(2) \) into the derivative formula:

\[ f'(2) = \lim_{x \to 2} \frac{f(x) - f(2)}{x - 2} = \lim_{x \to 2} \frac{f(x) - 0}{x - 2} = \lim_{x \to 2} \frac{f(x)}{x - 2} \]

### Step 3: Substitute \( f(x) \)

Next, we substitute \( f(x) = x^2 - 4 \):

\[ f'(2) = \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \]

### Step 4: Factor the numerator

Notice that \( x^2 - 4 \) can be factored as a difference of squares:

\[ x^2 - 4 = (x - 2)(x + 2) \]

So we can rewrite the limit:

\[ f'(2) = \lim_{x \to 2} \frac{(x - 2)(x + 2)}{x - 2} \]

### Step 5: Cancel the common terms

We can cancel \( x - 2 \) from the numerator and the denominator (as long as \( x \neq 2 \)):

\[ f'(2) = \lim_{x \to 2} (x + 2) \]

### Step 6: Evaluate the limit

Now we can directly substitute \( x = 2 \):

\[ f'(2) = 2 + 2 = 4 \]

### Final Answer

Thus, the value of \( f'(2) \) is:

\[ \boxed{4} \]