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

To find the instantaneous rate of change of the function \( f(x) = x^3 + 1 \) at \( x = 2 \) using the first principle (also known as the definition of the derivative), we use the following formula:

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

### Step 1: Substitute the function into the formula

Given \( f(x) = x^3 + 1 \), we substitute this into the formula for \( f'(x) \):

\[
f'(x) = \lim_{{h \to 0}} \frac{(x+h)^3 + 1 - (x^3 + 1)}{h}
\]

### Step 2: Simplify the expression

First, expand \( (x+h)^3 \):

\[
(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3
\]

Now substitute this back into the expression:

\[
f'(x) = \lim_{{h \to 0}} \frac{x^3 + 3x^2h + 3xh^2 + h^3 + 1 - x^3 - 1}{h}
\]

Simplify the numerator:

\[
f'(x) = \lim_{{h \to 0}} \frac{3x^2h + 3xh^2 + h^3}{h}
\]

Factor out \( h \) from the numerator:

\[
f'(x) = \lim_{{h \to 0}} \frac{h(3x^2 + 3xh + h^2)}{h}
\]

Cancel out the \( h \) in the numerator and denominator:

\[
f'(x) = \lim_{{h \to 0}} (3x^2 + 3xh + h^2)
\]

### Step 3: Evaluate the limit as \( h \) approaches 0

\[
f'(x) = 3x^2 + 3x(0) + 0^2 = 3x^2
\]

### Step 4: Find the derivative at \( x = 2 \)

Now, substitute \( x = 2 \) into the derivative:

\[
f'(2) = 3(2)^2 = 3 \times 4 = 12
\]

So, the instantaneous rate of change of \( f(x) = x^3 + 1 \) at \( x = 2 \) is \( 12 \).

Would you like more details or have any questions?

Here are five related questions to explore further:
1. How does the instantaneous rate of change compare with the average rate of change between two points?
2. How would you find the derivative of a more complex function using the first principle?
3. What is the geometric interpretation of the derivative at a specific point on the graph?
4. How does the first principle approach differ from using standard derivative rules?
5. Can you apply the first principle to find the derivative of \( f(x) = \sin(x) \) at a specific point?

**Tip:** The first principle method is foundational for understanding derivatives and can be applied to any differentiable function, making it essential to grasp before moving on to more advanced derivative rules.

Learn how to find the instantaneous rate of change of the function f(x) = x^3 + 1 at x = 2 using the first principle. This problem involves calculus concepts, including derivatives and the definition of the derivative formula. Suitable for advanced high school students.

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