The problem given is:

$\lim_{{h \to 0}} \frac{f(1+h) - f(1)}{h} \quad \text{where} \quad f(x) = 2x^3 - 2$

This is the definition of the derivative of $f(x)$ evaluated at $x = 1$. We will compute this step by step.

Step 1: Find $f(1+h)$

Given $f(x) = 2x^3 - 2$, substitute $x = 1 + h$:

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

Now expand $(1+h)^3$:

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

So,

$f(1+h) = 2(1 + 3h + 3h^2 + h^3) - 2$ $f(1+h) = 2 + 6h + 6h^2 + 2h^3 - 2$ $f(1+h) = 6h + 6h^2 + 2h^3$

Step 2: Find $f(1)$

Using $f(x) = 2x^3 - 2$, substitute $x = 1$:

$f(1) = 2(1)^3 - 2 = 2 - 2 = 0$

Step 3: Substitute into the difference quotient

Now substitute $f(1+h)$ and $f(1)$ into the expression for the limit:

$\frac{f(1+h) - f(1)}{h} = \frac{(6h + 6h^2 + 2h^3) - 0}{h} = \frac{6h + 6h^2 + 2h^3}{h}$

Simplify the expression:

$\frac{6h + 6h^2 + 2h^3}{h} = 6 + 6h + 2h^2$

Step 4: Take the limit as $h \to 0$

As $h \to 0$:

$6 + 6h + 2h^2 \to 6$

Therefore, the value of the limit is:

$\boxed{6}$

Let me know if you'd like further details on any of the steps.

Related Questions:

1. What is the general formula for the derivative using limits?
2. How do you expand a binomial expression like $(1+h)^3$?
3. How would you calculate the derivative for $f(x) = 2x^3 - 2$ directly using differentiation rules?
4. What are the conditions under which a limit fails to exist?
5. How can the derivative be interpreted geometrically?

Tip:

When computing limits, it's often helpful to simplify the expression first, especially when dealing with small values like $h \to 0$, as this can make the calculation easier.