Let f (x) = x2+5x+1. Find f ′(2) using the limit definition of the derivative

To find \( f'(2) \) using the limit definition of the derivative, we apply the formula:

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

In this case, \( f(x) = x^2 + 5x + 1 \), and we are asked to find \( f'(2) \). So we substitute \( x = 2 \) into the limit definition:

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

### Step 1: Calculate \( f(2+h) \)
\[
f(2+h) = (2+h)^2 + 5(2+h) + 1
\]
Expanding this:
\[
f(2+h) = (4 + 4h + h^2) + (10 + 5h) + 1
\]
\[
f(2+h) = h^2 + 9h + 15
\]

### Step 2: Calculate \( f(2) \)
\[
f(2) = 2^2 + 5(2) + 1 = 4 + 10 + 1 = 15
\]

### Step 3: Substitute into the limit formula
Now substitute these values into the limit definition:
\[
f'(2) = \lim_{h \to 0} \frac{(h^2 + 9h + 15) - 15}{h}
\]
Simplify the numerator:
\[
f'(2) = \lim_{h \to 0} \frac{h^2 + 9h}{h}
\]
Factor out \( h \) from the numerator:
\[
f'(2) = \lim_{h \to 0} \frac{h(h + 9)}{h}
\]
Cancel out \( h \) (for \( h \neq 0 \)):
\[
f'(2) = \lim_{h \to 0} (h + 9)
\]

### Step 4: Evaluate the limit
As \( h \to 0 \), we get:
\[
f'(2) = 9
\]

Thus, \( f'(2) = 9 \).

Would you like further details on any part of this process or have any questions?

Here are five related questions you might want to explore:

1. How do you find the derivative of \( f(x) = x^2 + 5x + 1 \) using standard differentiation rules?
2. Can you explain the physical meaning of the derivative at a point?
3. How does the limit definition of the derivative differ from simply applying differentiation rules?
4. What happens if we try to calculate \( f'(x) \) at a point where the function is not differentiable?
5. How would the calculation change if the function had a higher degree, such as \( f(x) = x^3 + 5x + 1 \)?

**Tip:** The limit definition of the derivative helps you understand differentiation from first principles, showing the derivative as the slope of the tangent line at a specific point.

Let f(x) = x^2 + 5x + 1. Find f ′(2) using the limit definition of the derivative.

Learn how to find the derivative f'(2) for the function f(x) = x^2 + 5x + 1 using the limit definition. This detailed step-by-step solution explores key calculus concepts like limits and derivatives. Ideal for high school students in Grades 11-12.

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