To solve this problem, we need the power rule, the derivative formulas for sine and cosine, and the chain rule.

The chain rule states that:

In this problem, we will have to apply the chain rule twice. This is because the  is inside the sine function which is inside the cosine function.

In this problem, , , and we have another function 

For this problem, we are using the chain rule in this form:

To evaluate these derivatives, we need the power rule and the derivatives of sine and cosine which state:

Now, plugging these equations into the chain rule, we obtain:

To solve this problem, we need the derivative of a constant, the derivative of the trigonometric function cosine, and the chain rule.

First, let's rewrite the function in terms of a power:

 Now we should apply the chain rule which states that:

In this problem,  and .

To find  we need to use the power rule, which states:

To find  , we again need to use the chain rule, the derivative of a constant, and the derivative of the rtigonometric function cosine to evaluate  , which state that:

Plugging all of these equations back into the chain rule, we obtain:

To solve this problem, we need the derivative of the trigonometric function cotangent, derivative of a constant, and the quotient rule.

First, let's use the quotient rule, which states:

In this problem,   and  .

To find , we need the formula for the derivative of cotangent which states:

To find  we also need the derivative of a constant formula which states:

Now combining these into the quotient rule formula, we obtain:

And after some simplification:

To solve this problem, we need the power rule, and the derivative of , which state:

After moving some things around with algebraic techniques, we obtain:

To solve this problem, we need the chain rule, the power rule, and the derivative of a constant.

Let's first rewrite the function in terms of a power:

Now we can use the chain rule, which states:

In this problem,  and 

To find  , we need the power rule which states:

To find , we need the power rule and the derivative of a constant which states:

Now, plugging these equations into the chain rule we obtain:

To solve this problem, we need the power rule and the chain rule, which state:

First let's apply the chain rule, which states:

In this problem,  and 

To find the , we need the power rule which states:

To find  we again need the power rule:

Now plugging these equations into the chain rule, we obtain:

To find the derivative of the function we must use the quotient rule.

It states that the derivative of 

 is .

The derivativae of  is  and the derivative of  is  as per the derivative rules.

Thus the final answer is 

The derivative is the function of the slope at any point of the given function. Thus we must find the derivative and then plug in 2 for x to get the slope of the tangent line.

The derivative of  is . The derivative of  is . So the derivative function is

plugging in  gives

.

The approach to this derivative is to realize that it is a function within a function and that we must use the chain rule.

The chain rule states the derivative of  is .

The derivative of  is  and the derivative of  is .

That makes the derivative of the function

.

To solve this problem, we need the power rule, the derivative of a constant, the product rule, and the chain rule.

Since our function is written as a product, we will first apply the product rule which states:

In this problem,  and .

To find  we need to use the power rule and the derivative of a constant which state that:

To find  we need to use the chain rule, the power rule, and the derivative of a constant. The chain rule states that:

In this problem,  and 

To find  we need the power rule.

To find  we need the power rule and the derivative of a constant once again.

Plugging these equations back into the chain rule, we obtain:

Now plugging this back into the product rule, we obtain:

After some simplification, we have: