Final answer:
To find the derivative of the function y = 1/2(x√ (16 -x²) + 16 arc sin (x/4)), apply the product rule, chain rule, and derivative rules for inverse trigonometric functions, and simplify.
Step-by-step explanation:
The derivative of the function y = 1/2(x√ (16 -x²) + 16 arc sin (x/4)) can be found using the product rule, the chain rule, and the derivative of the inverse trigonometric function. The product rule is applied to x√ (16 -x²), and the chain rule is used to differentiate √ (16 -x²). For the second part of the function, the derivative of arc sin (x/4) is found using knowledge of the derivatives of inverse trigonometric functions. The final derivative is a combination of these computed parts, simplified into a single expression.