Question

What is the derivative of the function x^3 + 2x^2 - 5x + 1?

Answer 1

To find the derivative of the function \( f(x) = x^3 + 2x^2 - 5x + 1 \), we can apply the power rule. The power rule states that if \( f(x) = ax^n \), then \( f'(x) = nax^{n-1} \).

Now, applying the power rule to each term:

1. The derivative of \( x^3 \) is \( 3x^2 \).
2. The derivative of \( 2x^2 \) is \( 2 \cdot 2x^{2-1} = 4x \).
3. The derivative of \( -5x \) is \( -5 \).
4. The derivative of the constant \( 1 \) is \( 0 \).

Combining these results, we get:

\[
f'(x) = 3x^2 + 4x - 5
\]

Thus, the derivative of the function \( x^3 + 2x^2 - 5x + 1 \) is:

\[
f'(x) = 3x^2 + 4x - 5
\]