Find the third derivative of the given function.
$f\left(x\right)=24+4x^5$

• A. $60x^2$
• B. $80x^3$
• C. $240x^2$
• D. $24-240x^2$

Answer: (C)

Verified step by step guidance

1. Start by identifying the function for which you need to find the third derivative. The function given is \( f(x) = 24 + 4x^5 \).
2. To find the third derivative, you first need to find the first derivative. Differentiate \( f(x) = 24 + 4x^5 \) with respect to \( x \). The derivative of a constant is zero, and the derivative of \( 4x^5 \) is \( 20x^4 \). So, the first derivative \( f'(x) = 20x^4 \).
3. Next, find the second derivative by differentiating \( f'(x) = 20x^4 \). The derivative of \( 20x^4 \) is \( 80x^3 \). Thus, the second derivative \( f''(x) = 80x^3 \).
4. Now, find the third derivative by differentiating \( f''(x) = 80x^3 \). The derivative of \( 80x^3 \) is \( 240x^2 \). Therefore, the third derivative \( f'''(x) = 240x^2 \).
5. Verify your result by checking each differentiation step to ensure accuracy. The third derivative of the function \( f(x) = 24 + 4x^5 \) is \( 240x^2 \).