To solve this problem, we need to use the quotient rule, chain rule, derivative of the cosine function, the derivative of a constant, and the power rule.

Let's recall the quotient rule:

In this problem,   and 

To find the derivative of , we need to use the derivative of cosine:

And the chain rule: 

Where  and 

Therefore  and 

Combining these terms, we have the derivative of our numerator:

Now to find the derivative of the denominator, we will need the power rule and the derivative of a constant:

Using these formulas, 

Now plugging these quantities into the quotient rule, we obtain:

Now, multiplying all of these terms together and simplifying with some algebra, you obtain:

The first thing that we should do with this problem is rewrite the function in terms of powers, so that we can use the power rule:

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

First, we should apply the chain rule, which states:

Where  and .

To evaluate , we need the power rule which states:

To find , we also need to use the power rule:

Plugging these values into the chain rule, we obtain:

To solve this problem, we must first evalute the derivative and then plug in  .

To find this derivative, we need the derivative of the tangent function:

Now, we plug in  for  in this derivative to obtain:

Therefore, the derivative does not exist at this point.

To solve this problem, we first need to take the derivative of the function. Then, we will set the derivative equal to  to find the  value where the slope of the tangent line is equal to . Finally, we will plug this  back into the original function to find the corresponding  value. 

We will need the power rule and the derivative of a constant to solve this problem:

Now we set the derivative equal to :

Now that we have the  value where the slope of the tangent line is equal to , we plug that value back into the function to find the  value:

Therefore, the point where the slope of the tangent line is equal to  is 

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: