Differentiate f(x) = x^3 + 2x using the definition of the derivative.

To differentiate f(x) = x^3 + 2x using the definition of the derivative, we need to find the limit of the difference quotient as h approaches 0.

The difference quotient is given by:

f'(x) = lim(h->0) [f(x + h) - f(x)] / h

Substitute f(x) = x^3 + 2x into the formula:

f'(x) = lim(h->0) [(x + h)^3 + 2(x + h) - (x^3 + 2x)] / h
f'(x) = lim(h->0) [x^3 + 3x^2h + 3xh^2 + h^3 + 2x + 2h - x^3 - 2x] / h
f'(x) = lim(h->0) [3x^2h + 3xh^2 + h^3 + 2h] / h
f'(x) = lim(h->0) 3x^2 + 3xh + h^2 + 2
f'(x) = 3x^2 + 2

Therefore, the derivative of f(x) = x^3 + 2x is f'(x) = 3x^2 + 2.

Definition of the Derivative:
The derivative of a function f(x) at a point x is defined as:

```
f'(x) = lim(h->0) [f(x+h) - f(x)] / h
```

Applying the Definition:
Using this definition, we can calculate the derivative of f(x) = x^3 - 2x:

```
f'(x) = lim(h->0) [((x+h)^3 - 2(x+h)) - (x^3 - 2x)] / h
```

Simplifying the Expression:

```
f'(x) = lim(h->0) [(x^3 + 3x^2h + 3xh^2 + h^3 - 2x - 2h) - (x^3 - 2x)] / h
```

```
f'(x) = lim(h->0) [3x^2h + 3xh^2 + h^3 - 2h] / h
```

Canceling out h:

```
f'(x) = lim(h->0) [3x^2 + 3xh + h^2 - 2] / h
```

```
f'(x) = lim(h->0) [3x^2 + 3xh + h^2] / h - lim(h->0) [2] / h
```

Evaluating the Limits:

```
f'(x) = 3x^2 + 3x - 0 + 0^2 - 2 - 0
```

```
f'(x) = 3x^2 - 2
```

Therefore, the derivative of f(x) = x^3 - 2x is f'(x) = 3x^2 - 2.