Final answer:
The student is seeking the derivative of a complex function at the point x=1. The calculation involves applying the chain rule, product rule, and properties of inverse trigonometric functions. Due to ambiguity in the function's notation, exact computation is not provided without clarification.
Step-by-step explanation:
The student is asking to find the derivative of the function [cos⁻¹ (sin √1 +x / 2) + xˣ] concerning x at x=1. To find the derivative, we need to apply the chain rule, product rule, and the properties of the inverse trigonometric functions. However, based on the information provided, the function has some ambiguities, particularly the term √1 +x / 2, where the expression under the square root is unclear due to the missing parenthesis. Assuming the correct interpretation, the chain rule and the derivatives of the inverse cosine and sine functions would be used to compute the derivative at x=1. It is essential here to remember that the sign of the inverse trigonometric functions is limited by their range and that the function xˣ has a derivative that includes a term with a natural logarithm.
As the exact function is not fully clear and there could be typos, the exact derivative calculation cannot be provided without clarification. Since the derivative is a fundamental topic in calculus, it is vital to ensure that the function is correctly written before attempting differentiation.