Final answer:
To find the 31st derivative of f(x) = (x⁶-x⁴)⁵ at x=0, the chain rule is applied repeatedly, resulting in a final answer of 0.
Step-by-step explanation:
To find the 31st derivative of f(x) = (x⁶-x⁴)⁵ at x=0, we can use the chain rule. The derivative of (x⁶-x⁴)⁵ is given by:
f'(x) = 5(x⁶-x⁴)⁴(6x⁵-4x³)
Continuing this process, we can find the 31st derivative by repeatedly applying the chain rule:
f''(x) = 20(x⁶-x⁴)³(36x¹⁰-12x⁸)
...
f⁽³⁰⁾(x) = 5(6x¹⁸-4x¹⁶)
f⁽³¹⁾(x) = 0
Therefore, the 31st derivative of f(x) at x=0 is 0. Answer choice A is correct.