Question

If $f(x) = x^2 + 3x$, then $f'(x)$ is:

• A. $2x + 3$
• B. $x + 3$
• C. $2x$
• D. $x^2$

Hint:

The derivative of any linear term $ax$ is simply the coefficient $a$. The derivative of a constant is always $0$.

Answer 1

The Correct Option is A

To find the derivative of the function \( f(x) = x^2 + 3x \), we need to apply the basic rules of differentiation.

1. Identify the function \( f(x) = x^2 + 3x \).
2. Differentiate each term separately using the power rule for derivatives. The power rule states that if \( f(x) = x^n \), then its derivative \( f'(x) = nx^{n-1} \).

Let's differentiate \( f(x) = x^2 \):

• Using the power rule, the derivative of \( x^2 \) is \( 2x \).

Now, differentiate \( f(x) = 3x \):

• The derivative of \( 3x \) is simply the coefficient of \( x \), which is \( 3 \).

Combine the derivatives of both terms:

• \( f'(x) = 2x + 3 \).

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

This matches the correct answer given as $2x + 3$. The other options are incorrect because they do not satisfy the derivative calculation:

• $x + 3$: Incorrect as it does not include the \( 2x \) term from the derivative of \( x^2 \).
• $2x$: Incorrect because it lacks the constant term 3, which comes from differentiating \( 3x \).
• $x^2$: Incorrect as it simply repeats the original function, not the derivative.

Therefore, the correct answer is indeed $2x + 3$.