Final answer:
The derivative of Y=cos(x^2+8) using the chain rule is -sin(x^2+8) × 2x.
Step-by-step explanation:
The student has asked how to differentiate the function Y=cos(x^2+8) using the chain rule. To apply the chain rule, we recognize that there are two functions involved: the outer function which is the cosine function and the inner function which is x^2 + 8. When differentiating Y with respect to x, we first differentiate the outer function with respect to the inner function, and then multiply by the derivative of the inner function with respect to x.
Therefore, the derivative of Y with respect to x is given by:
dY/dx = -sin(x^2+8) × (2x).
This result incorporates the derivative of the cosine function, which is -sin( ), and the derivative of x^2 which is 2x, applying the chain rule correctly.