Final answer:
The derivative of f(x) = (5 - x²)/(5x²) is found by applying the quotient rule. After calculating and simplifying, the correct derivative is f'(x) = -2 / (5x³), and none of the given choices are correct. The right answer is f'(x) = -2 / (5x³).
Step-by-step explanation:
The question asks us to find the derivative of the function f(x) = (5 - x²)/(5x²). We can solve this by using the quotient rule, which states that the derivative of a function that is the quotient of two functions, u(x) and v(x), is given by:
(u'v - uv') / v²
where u' and v' are the derivatives of u and v, respectively.
In this case, u(x) = 5 - x² and v(x) = 5x². The derivatives u' and v' will be:
Applying the quotient rule:
f'(x) = ((-2x)(5x²) - (5 - x²)(10x)) / (5x²)²
Simplify and combine like terms:
f'(x) = (-10x³ - 50x + 10x³) / (25x´)
f'(x) = (-50x) / (25x´)
After simplifying:
f'(x) = -2 / (5x³)
None of the given options are correct. The right answer is f'(x) = -2 / (5x³).