Find the instantaneous rate of change of 𝑓(𝑥) = 𝑥3 + 1 at 𝑥 = 2 using the first principle

The instantaneous rate of change of a function at a given point is the derivative of the function at that point. According to the first principle of derivatives, the derivative of \( f(x) \) at \( x = a \) is given by:

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

For the function \( f(x) = x^3 + 1 \), we want to find the derivative at \( x = 2 \).

### Step 1: Compute \( f(2 + h) \)
Substitute \( x = 2 + h \) into \( f(x) \):

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

### Step 2: Expand \( (2 + h)^3 \)
\[
(2 + h)^3 = 8 + 12h + 6h^2 + h^3
\]

So,
\[
f(2 + h) = 8 + 12h + 6h^2 + h^3 + 1 = 9 + 12h + 6h^2 + h^3
\]

### Step 3: Compute \( f(2 + h) - f(2) \)
\( f(2) = 2^3 + 1 = 8 + 1 = 9 \), so

\[
f(2 + h) - f(2) = (9 + 12h + 6h^2 + h^3) - 9 = 12h + 6h^2 + h^3
\]

### Step 4: Compute the difference quotient
\[
\frac{f(2 + h) - f(2)}{h} = \frac{12h + 6h^2 + h^3}{h} = 12 + 6h + h^2
\]

### Step 5: Take the limit as \( h \to 0 \)
\[
f'(2) = \lim_{{h \to 0}} (12 + 6h + h^2) = 12
\]

### Final Answer:
The instantaneous rate of change of \( f(x) = x^3 + 1 \) at \( x = 2 \) is \( \boxed{12} \).

Would you like further details or have any questions?

Here are some related questions to explore:
1. How do you find the derivative of \( f(x) = x^3 \) using the power rule?
2. What is the significance of the first principle in calculus?
3. Can you apply the first principle to find the derivative of \( f(x) = \frac{1}{x} \) at a specific point?
4. How do you interpret the derivative in the context of a real-world problem?
5. What is the difference between average and instantaneous rates of change?

**Tip:** Always simplify the expression before taking the limit to make the calculation easier!

Learn how to find the instantaneous rate of change of the function f(x) = x^3 + 1 at x = 2 using the first principle of derivatives. This problem involves calculating the derivative of a cubic function and applying the limit definition to find the rate of change. Suitable for advanced high school students and calculus enthusiasts.

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