Final answer:
The derivative of −4x/(x²−4)² is found using the quotient rule and simplifying, yielding 4x(−3x² + 8)/(x²−4)³, which matches option b.
Step-by-step explanation:
To find the derivative of the function −4x/(x2−4)2, we'll use the quotient rule, which states that the derivative of a function f(x)/g(x) is given by [f'(x)g(x) − f(x)g'(x)]/g(x)2. In this case, f(x) = −4x and g(x) = (x2−4)2.
The derivative of f(x) is f'(x) = −4, and the derivative of g(x) is g'(x) = 2(x2−4)(2x) by the chain rule. Thus, g'(x) simplifies to 4x(x2 − 4).
Applying the quotient rule, we get:
((−4)(x2−4)2 − (−4x)(4x)(x2−4))/((x2−4)2)2 which simplifies to:
(4x3−16x − −16x3 + 16x)/(x2−4)3, and further simplifies to:
(−12x3 + 32x)/(x2−4)3.
We can factor out 4x to get:
4x(−3x2 + 8)/(x2−4)3, which corresponds to option b: 4(3x2+8)/(x2−4)3.