Final answer:
To find df(x)/dx for f(x) = cos⁻¹(x) + cos⁻¹{2 + (2/3) - 3x²}, one must apply the chain rule and the derivative of the inverse cosine function carefully considering their domains.
Step-by-step explanation:
The question involves finding the derivative of the function f(x) = cos⁻¹(x) + cos⁻¹{2 + (2/3) - 3x²}. To differentiate this function with respect to x, we will need to apply the chain rule and the derivative of the inverse cosine function. Remember that the derivative of cos⁻¹(u) with respect to u is -1/√(1-u²). So, if we let u = 2 + (2/3) - 3x², we have:
du/dx = -6xdf(x)/dx = -1/√(1-x²) - (1/√(1-u²))(-6x)
However, this expression must be evaluated with attention to the domain of the inverse cosine function, where x must be between -1 and 1, and 2 + (2/3) - 3x² must also lie within this interval.