To find the value after differentiating cos(sin(x)), we need to apply the chain rule. The chain rule states that if we have a composite function, F(g(x)), then the derivative of F with respect to x is given by F'(g(x)) * g'(x).

The given function cos(sin(x)) is a composite function where the outer function is cos(x) and the inner function is sin(x). So, we can apply the chain rule to differentiate it.

Let's break down the steps to differentiate cos(sin(x)):

Step 1: Identify the Inner and Outer Functions
The outer function is cos(x) and the inner function is sin(x).

Step 2: Find the Derivative of the Inner Function
The derivative of sin(x) with respect to x is cos(x). This is because the derivative of sin(x) is cos(x).

Step 3: Find the Derivative of the Outer Function
The derivative of cos(x) with respect to x is -sin(x). This is the basic derivative of cos(x).

Step 4: Apply the Chain Rule
According to the chain rule, we need to multiply the derivative of the outer function by the derivative of the inner function.

So, the derivative of cos(sin(x)) is:
- sin(sin(x)) * cos(x)

Answer: Option (b) -sin(sin(x)) * cos(x)

In summary, to find the value after differentiating cos(sin(x)), we applied the chain rule. The derivative of the outer function cos(x) is -sin(x), and the derivative of the inner function sin(x) is cos(x). Multiplying these two derivatives gives us -sin(sin(x)) * cos(x).