To find the instantaneous rate of change of $f(x) = x^3 + 1$ at $x = 2$ using the first principle of differentiation, we follow these steps:

First Principle of Derivatives:

The instantaneous rate of change of a function $f(x)$ at $x = a$ is given by the limit:

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

Given Function:

$f(x) = x^3 + 1$

We want to find the rate of change at $x = 2$. So, $a = 2$, and we apply the formula:

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

Step-by-Step Calculation:

1. Find $f(2+h)$:

$f(2+h) = (2+h)^3 + 1$

Expand $(2+h)^3$:

$(2+h)^3 = 8 + 12h + 6h^2 + h^3$

So,

$f(2+h) = 8 + 12h + 6h^2 + h^3 + 1 = 9 + 12h + 6h^2 + h^3$

2. Find $f(2)$:

$f(2) = 2^3 + 1 = 8 + 1 = 9$

3. Substitute into the limit expression:

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

Simplify the numerator:

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

4. Simplify the fraction:

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

5. Evaluate the limit as $h \to 0$:

$f'(2) = 12$

Conclusion:

The instantaneous rate of change of $f(x) = x^3 + 1$ at $x = 2$ is $12$.

---

Would you like further details on any step, or do you have any questions?

Related Questions:

1. What is the first principle of differentiation for other types of functions?
2. How does the first principle relate to the concept of derivatives in general?
3. Can we find the rate of change of a trigonometric function using the first principle?
4. What is the significance of the instantaneous rate of change in real-life applications?
5. How can we apply the first principle to a function with more complex powers?

Tip:

When using the first principle of differentiation, always simplify the expressions as much as possible before applying the limit!