The given problem asks to differentiate the function $f(x) = x^5 - x^3 + 2x$ and evaluate the derivative at $x = 2$.

Step 1: Differentiate the function

Using standard differentiation rules:

1. The derivative of $x^n$ is $nx^{n-1}$.
2. The derivative of $2x$ is $2$.

$f'(x) = \frac{d}{dx}(x^5) - \frac{d}{dx}(x^3) + \frac{d}{dx}(2x)$

$f'(x) = 5x^4 - 3x^2 + 2$

Step 2: Evaluate $f'(x)$ at $x = 2$

Substitute $x = 2$ into $f'(x)$:

$f'(2) = 5(2^4) - 3(2^2) + 2$

$f'(2) = 5(16) - 3(4) + 2$

$f'(2) = 80 - 12 + 2$

$f'(2) = 70$

Final Answer:

The derivative of $f(x)$ at $x = 2$ is $\mathbf{70}$.

Would you like more details on any of the steps? Here are some related questions to explore further:

1. What are the general rules of differentiation, such as the power rule?
2. How do you handle higher-order derivatives for a polynomial function?
3. Can we integrate the derivative to recover the original function?
4. How does this derivative behave as $x$ approaches infinity?
5. How does the derivative's sign indicate whether the function is increasing or decreasing?

Tip: Always simplify your derivative fully before substituting values to avoid computational errors!