Final answer:
The derivative of the function f(x) = ∛(4x⁴ + x² - 1)⁵ can be found using the chain rule and power rule. The result is f'(x) = 5∛(4x⁴ + x² - 1)⁴ ∙ (16x³ + 2x).
Step-by-step explanation:
To find the derivative of the function f(x) = ∛(4x⁴ + x² - 1)⁵, we need to use the chain rule combined with the power rule and the properties of derivatives. This process is often referred to as taking the outer derivative and then the inner derivative.
Let's first let u = (4x⁴ + x² - 1), then f(x) becomes (∛u)⁵. The derivative of f with respect to x is given by applying the chain rule:
d/dx [(u)⁵] ∙ du/dx.
For u = (4x⁴ + x² - 1), the derivative du/dx is 16x³ + 2x. Then we apply the power rule to (∛u)⁵, treating u as the variable, and multiply it by the derivative of u, yielding:
(5∛u⁴) ∙ (16x³ + 2x).
Finally, we substitute u back in to get the derivative of f with respect to x:
f'(x) = 5∛(4x⁴ + x² - 1)⁴ ∙ (16x³ + 2x).