Answer :
Final Answer
For the function s(x) = 20x³ - 3x⁵ :
(i) s'(x) = 60x² - 15x⁴, (ii) s''(x) = 120x - 60x³
Thus the correct option is A
Explanation
We need to find the first and second derivatives of the function
s(x) = 20x³ - 3x⁵. Here's how to solve it:
First derivative (s'(x)):
Apply the power rule:
d/dx(x^n) = nx^(n-1).
s'(x) = d/dx(20x³) - d/dx(3x⁵)
= (3 * 20x²) - (5 * 3x⁴) = 60x² - 15x⁴
Second derivative (s''(x)):
Differentiate s'(x) which is a polynomial.
s''(x) = d/dx(60x²) - d/dx(15x⁴)
= (2 * 60x) - (4 * 15x³) = 120x - 60x³
Therefore, the first derivative is
s'(x) = 60x² - 15x⁴ and the second derivative is
s''(x) = 120x - 60x³.
