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 (definition of the derivative), we follow these steps:

### Step 1: Definition of the Derivative
The derivative of \( f(x) \) at a point \( x = a \) is given by:

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

Here, \( a = 2 \). So, we need to find:

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

### Step 2: Calculate \( f(2+h) \) and \( f(2) \)
First, we find \( f(2) \):

\[
f(2) = 2^3 + 1 = 8 + 1 = 9
\]

Next, we calculate \( f(2+h) \):

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

Expand \( (2+h)^3 \) using the binomial theorem:

\[
(2+h)^3 = 8 + 12h + 6h^2 + h^3
\]

So,

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

### Step 3: Set Up the Difference Quotient
Now, substitute \( f(2+h) \) and \( f(2) \) into the difference quotient:

\[
f'(2) = \lim_{{h \to 0}} \frac{{(9 + 12h + 6h^2 + h^3) - 9}}{h}
\]

Simplify the expression:

\[
f'(2) = \lim_{{h \to 0}} \frac{{12h + 6h^2 + h^3}}{h}
\]

### Step 4: Simplify and Take the Limit
Factor out \( h \) from the numerator:

\[
f'(2) = \lim_{{h \to 0}} \frac{h(12 + 6h + h^2)}{h}
\]

Cancel out \( h \) (as long as \( h \neq 0 \)):

\[
f'(2) = \lim_{{h \to 0}} (12 + 6h + h^2)
\]

Finally, take the limit as \( h \) approaches 0:

\[
f'(2) = 12
\]

### Conclusion
The instantaneous rate of change of the function \( f(x) = x^3 + 1 \) at \( x = 2 \) is 12.

Would you like more details or have any questions? Here are some related questions:

1. How do you find the derivative using the power rule?
2. What is the geometric interpretation of the instantaneous rate of change?
3. How does the concept of the limit relate to the derivative?
4. What happens if you apply the first principle at a different point?
5. Can the first principle be applied to any type of function?

**Tip:** Understanding how to expand binomials and simplify expressions is crucial when applying the first principle to find derivatives.

Learn how to calculate the instantaneous rate of change of the function f(x) = x^3 + 1 at x = 2 using the first principle (definition of the derivative). Follow a step-by-step solution involving the definition of the derivative, difference quotient, and simplification techniques. Suitable for advanced high school students.

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