Final answer:
To find the derivative of the function f(x) = x³ / (1 - 4x⁸), we apply the quotient rule and find that f'(x) = (3x²(1 - 4x⁸) - x³(-32x⁷))/(1 - 4x⁸)². We then simplify the expression to obtain the final derivative.
Step-by-step explanation:
The student has asked to find the derivative of the function f(x) = x³ / (1 - 4x⁸). This requires the application of the quotient rule for derivatives since the function is given as the quotient of two functions. The quotient rule is expressed as (f(x)/g(x))' = (f'(x)g(x) - f(x)g'(x))/(g(x))^2. In this case, let u = x³ and v = 1 - 4x⁸. The derivatives u' = 3x² and v' = -32x⁷. Applying the quotient rule:
(f(x)/g(x))' = (3x²(1 - 4x⁸) - x³(-32x⁷))/(1 - 4x⁸)²
Simplify the expression to get the final result.
Note that the typos and irrelevant information provided in the question do not affect the process of finding the derivative using the quotient rule, and the details provided seem to be from different problems or contexts, unrelated to the main problem.
To find the derivative of the function f(x) = x³ / (1 - 4x⁸), we can use the quotient rule. The quotient rule states that if we have a function f(x) = g(x) / h(x), the derivative of f(x) is given by f'(x) = [g'(x) * h(x) - g(x) * h'(x)] / [h(x)]².
Let's apply the quotient rule to the given function:
f'(x) = [(3x² * (1 - 4x⁸)) - (x³ * (-32x⁷))] / (1 - 4x⁸)²
Simplifying further, we get:
f'(x) = (3x² - 32x¹⁰) / (1 - 4x⁸)²