for f(x) = x^2 + 3x + 1 find f'(1) using f'(a)= lim x->a (f(x)-f(a))/(x-a)

To find \( f'(1) \) using the definition of the derivative, which is:

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

In this case, \( a = 1 \), so we need to compute \( f'(1) \). The function is \( f(x) = x^2 + 3x + 1 \), and we proceed with the following steps:

### Step 1: Compute \( f(1) \)
We first evaluate \( f(1) \):

\[
f(1) = (1)^2 + 3(1) + 1 = 1 + 3 + 1 = 5
\]

### Step 2: Write the difference quotient
Now, we use the difference quotient formula:

\[
f'(1) = \lim_{x \to 1} \frac{f(x) - f(1)}{x - 1}
\]

Substitute \( f(x) = x^2 + 3x + 1 \) and \( f(1) = 5 \):

\[
f'(1) = \lim_{x \to 1} \frac{(x^2 + 3x + 1) - 5}{x - 1}
\]

### Step 3: Simplify the expression
Simplify the numerator:

\[
f'(1) = \lim_{x \to 1} \frac{x^2 + 3x + 1 - 5}{x - 1} = \lim_{x \to 1} \frac{x^2 + 3x - 4}{x - 1}
\]

Now, factor the quadratic \( x^2 + 3x - 4 \):

\[
x^2 + 3x - 4 = (x - 1)(x + 4)
\]

So the expression becomes:

\[
f'(1) = \lim_{x \to 1} \frac{(x - 1)(x + 4)}{x - 1}
\]

### Step 4: Cancel common terms
Cancel the \( (x - 1) \) terms:

\[
f'(1) = \lim_{x \to 1} (x + 4)
\]

### Step 5: Evaluate the limit
Now, evaluate the limit as \( x \to 1 \):

\[
f'(1) = 1 + 4 = 5
\]

Thus, \( f'(1) = 5 \).

Would you like further clarification, or details on any of the steps? 

Here are five related questions to deepen your understanding:

1. How would you compute \( f'(a) \) for any general value of \( a \) using this same process?
2. How does the result compare with using basic derivative rules like the power rule for this problem?
3. What happens when a quadratic function is used in the limit definition of the derivative?
4. How would you approach a similar problem if the function were cubic instead of quadratic?
5. Can you think of a case where factoring might not work directly, and how could you handle it?

**Tip:** Always try simplifying the numerator in a difference quotient by factoring to cancel common terms with the denominator.

For f(x) = x^2 + 3x + 1 find f'(1) using f'(a)= lim x->a (f(x)-f(a))/(x-a)

Learn how to compute f'(1) for the function f(x) = x^2 + 3x + 1 using the limit definition of the derivative. This step-by-step solution covers key calculus concepts such as limits and derivatives, suitable for high school students in Grades 11-12.

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