Question

What is the derivative of f (x) = cosine Superscript negative 1 Baseline StartRoot 1 minus x Superscript 4 Baseline EndRoot?

A) f prime (x) = StartFraction negative 4 x cubed Over StartRoot 1 minus x Superscript 4 Baseline EndRoot EndFraction
B) f prime (x) = StartFraction negative 2 x Over StartRoot 1 minus x Superscript 4 Baseline EndRoot EndFraction
C) f prime (x) = StartFraction 2 x Over StartRoot 1 minus x Superscript 4 Baseline EndRoot EndFraction
D) f prime (x) = StartFraction 4 x cubed Over StartRoot 1 minus x Superscript 4 Baseline EndRoot EndFraction

Answer 1

The correct option is D, where the derivative of the function f(x) is calculated using the chain rule applied to the inverse cosine function and its nested radical function.

Step-by-step explanation:

The question involves finding the derivative of the inverse cosine function with a nested radical expression. To find the derivative of f(x) = cos-1(√(1 - x4)), we will use the chain rule and the derivative of the inverse cosine function. First, let's denote the inside function as g(x) = √(1 - x4).

The derivative of the inverse cosine function, cos-1(u), concerning u is -1/√(1-u2). Now apply the chain rule:

f'(x) = d/dx [ cos-1(g(x)) ]
= -1/√(1 - g(x)2) × g'(x)
= -1/√(1 - (√(1 - x4))2) × d/dx [√(1 - x4)]
= -1/√(1 - (1 - x4)) × (1/2)(1 - x4)-1/2 × -4x3
= 2x3√(1 - x4)-1
= StartFraction 2 x3 Over √(1 - x4) EndFraction
Thus, the correct option is D) f'(x) = StartFraction 4 x3 Over √(1 - x4) EndFraction.