Find the derivative of the function. y = cot2(sin(θ))

Answer 1

In order to solve this problem we need to use the chain rule :)

Remember that according to the chain rule, to find the derivative of f(g(x)) with respect to x, we multiply:

f'(g(x)) * g'(x)

where:

1. f(x) and g(x) are just regular functions
2. f'(x) is the derivative of f with respect to x
3. g'(x) is the derivative of g with respect to x

So to restate, the chain rule is:

d/dx [f(g(x))] = f'(g(x)) * g'(x)

In this example, we can say that

1. f(x) = cot(x)
2. g(x) = 2 * sin(x)
3. f(g(x)) = cot (2 * sin(x))

Using the chain rule, we know that the derivative of f(g(x)) = f'(g(x)) * g'(x).

1. First calculate g'(x) = [d/dx] 2 * sin(x) = 2 * cos(x)
2. Then calculate f'(g(x)) = -csc²( 2 * sin(x))
3. Now multiply them together: f'(g(x)) * g'(x) = -csc²( 2 * sin(x)) * 2 * cos(x)

So our answer is:

-2 * csc²(2sin(x)) * cos(x)

Hope this helps!