Final answer:
The derivative of the function y = cos(x) - cos(x)³ is found using the chain rule and power rule of differentiation to be -sin(x) + 3sin³(x), which matches option b).
Step-by-step explanation:
The question asks us to find the derivative of the function y = cos(x) - cos(x)³ using the rules of differentiation. To do this, we need to apply the chain rule and the power rule of differentiation.
First, let's differentiate y = cos(x). The derivative of cos(x) with respect to x is -sin(x), according to basic differentiation rules.
Next, differentiate the term -cos(x)³. To apply the chain rule, take the outside function which is the cube and differentiate that first, which gives us 3 times the inside function squared, so -3cos²(x). Then, differentiate the inside function cos(x), which gives us -sin(x), following the basic differentiation rules.
Combining these, the derivative becomes: -sin(x) - 3cos²(x)(-sin(x)), which simplifies to -sin(x) + 3sin(x)cos²(x). Using a trigonometric identity sin²(x) = 1 - cos²(x), we can rewrite cos²(x) as 1 - sin²(x), and thus the final derivative is:
-sin(x) + 3sin(x)(1 - sin²(x)) = -sin(x) + 3sin(x) - 3sin³(x)
After simplifying, the final answer is: -sin(x) + 3sin³(x), which corresponds to option b).