Let's examine the image closely:

The function given is: $f(x) = 2x^3 - x^2 + 3x$ The problem asks for the derivative of $f(x)$, denoted by $f'(x)$.

The derivative provided is: $f'(x) = 6x^2 - 2x + 3$

Now, let's compute the derivative to verify the result:

• The derivative of $2x^3$ is $6x^2$.
• The derivative of $-x^2$ is $-2x$.
• The derivative of $3x$ is $3$.

So the correct derivative is indeed: $f'(x) = 6x^2 - 2x + 3$

Everything in the solution is correct.

Would you like a more detailed explanation of how the derivatives were computed, or do you have any other questions?

Here are some related questions you might want to explore:

1. How do you find higher-order derivatives of this function?
2. What is the interpretation of the derivative $f'(x)$ in practical terms?
3. Can you find the critical points of this function by solving $f'(x) = 0$?
4. How do you determine the concavity of the graph of $f(x)$?
5. What is the equation of the tangent line to the graph at a particular point?

Tip: Always check each term's power when differentiating polynomial functions to avoid simple mistakes.