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

The first thing that we should do is to change the form of the function so that it is written as a power:

Now that it is written as a power, we can use the power rule and chain rule:

Recall these derivative formulas:

In this problem,  and 

 and

Now, combining these two derivatives with multiplication as demonstrated by the chain rule yields:

In using implicit differentiation, we need the power rule, the product rule, the chain rule, and the derivative of a constant, and the derivative of the trigonometric function cosine. 

To find the derivative of , we first need the product rule:

In this problem,  and 

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

To find the derivative of , we need the chain rule and the derivative of the trigonometric function cosine which state:

Where  and 

 and  

Combining these results with multiplication as demonstrated by the chain rule yields:

Now that we have  and , we can use the product rule:

Now, some algebraic simplification:

To find , we must use the power rule, the derivative of a consant, and the derivative of y.

Then using some algebraic techniques to solve for   :

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

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

 In this problem,  and 

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

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

First, let's write  as a power:

Using the chain rule with  and  we obtain:

Using the chain rule, 

Now that we have found  and , we can plug these values into the product rule, to obtain:

To find this derivative, we need the power rule and the quotient rule.

Using the quotient rule, which states:

In this problem,  and 

To find, we need the power rule which states:

And to find , we also need the power rule.

 Now, applying the quotient rule, we obtain:

Then we use algebraic methods to simplify the derivative:

To solve this problem, we need the product rule, and the derivatives of the trigonometric functions sine and cotangent.

First, we apply the product rule, which states:

In this problem.  and .

To find , we need the formula for the derivative of sine:

To find the  , we need the derivative of cotangent:

Now, plugging these values into the product rule, we obtain:

And after some simplification:

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: