Final answer:
The question relates to differentiating a composite function using the chain rule. The correct derivative of cos(sin(5θ)) involves taking the derivative of the cosine function, followed by multiplying with the derivative of the inner sine function.
Step-by-step explanation:
The question pertains to the application of the chain rule in calculus for differentiating composite functions. There seems to be some confusion with the notation and application of this rule in the original question, specifically when attempting to find dθ/d(cos(sin(5θ))). To clarify, the chain rule dictates that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
When applying the chain rule to the function cos(sin(5θ)), we must first differentiate the outer function, which in this case is the cosine function, resulting in -sin(sin(5θ)). Then, we multiply this by the derivative of the inner function, which is the derivative of sin(5θ) with respect to θ, giving us 5cos(5θ). Therefore, the correct application of the chain rule gives us the derivative dθ/d(cos(sin(5θ))) = -sin(sin(5θ)) · 5cos(5θ).
In the context of trigonometry, this process could be related to identities such as sin(2α) = 2sin(α)cos(α) or the laws like the Law of Sines and the Law of Cosines, which can be used for solving triangles.