Final answer:
To find the derivative of fx = cos^-1(4x^3), apply the chain rule, with the inner function g(x) = 4x^3. The derivative is then f'(x) = -12x^2/sqrt(1-16x^6).
Step-by-step explanation:
To find the derivative of the function fx = cos-1(4x3), we will use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
First, let's denote the inner function as g(x) = 4x3. The derivative of the inverse cosine function, cos-1(u), with respect to u is -1/√(1-u2). Since the inner function g(x) is 4x3, we find its derivative g'(x) = 12x2.
Applying the chain rule, the derivative of f(x) with respect to x is:
f'(x) = -1/√(1-(4x3)2) × 12x2
Now simplify the expression:
f'(x) = -12x2/√(1-16x6)