Question

Compute the derivative of the given function. 32. y= √sin(cos² x)​

Answer 1

To compute the derivative of y = √sin(cos²x), apply the chain rule and differentiate the outer function, the inner function, and the exponent function. The derivative is dy/dx = (1/2√sin(cos²x)) * (cos(sin(cos²x))) * (-sin(2cos²x)).

Step-by-step explanation:

To compute the derivative of the given function, y = √sin(cos²x), we can use the chain rule. Let's break it down step by step:

1. Start by differentiating the outer function, which is √u. The derivative of √u is (1/2√u) times the derivative of u.
2. Next, differentiate the inner function, u = sin(cos²x). The derivative of sin(u) is cos(u) times the derivative of u.
3. Finally, find the derivative of the exponent function, u = cos²x. The derivative of cos²u is -sin(2u) times the derivative of u.

Combining these steps, we get the derivative of y as: dy/dx = ((1/2√sin(cos²x)) * (cos(sin(cos²x))) * (-sin(2cos²x)))