Differentiate both sides with respect to :

By the sum rule:

By the chain rule:

Applying some algebra:

Differentiate both sides with respect to :

Apply the sum, difference, and constant multiple rules:

In the first term, apply the chain rule; in the second, apply the constant multiple rule:

Apply the power rule:

Now apply some algebra:

So now this is a three layer chain rule differentiation. The more functions combine to form the composite function the harder it will be to keep track of the derivative. I find it helpful to lay out each equation and each derivative, so:

Then a three layer chain rule is just the same as a two layer, except... there's one more layer!

It is still the outermost layer evaluated at the inner layers, and then move another layer in and repeat

So with implicit differentiation, you are going to be taking the derivative of every variable, in the entire equation. Every time you take the derivative of a variable, you have it's rate of change multiplied on the right. In this case, dx or dy. 

The result of the derivative is:

The first step is to create the  term that you are looking to solve for. This is done by dividing the entire equation by  to get it on the bottom of the fraction. After distributing this division to each term, the dx in the first term will cancel with itself, and you will be left with one term that is multiplied by . At that point, you want to get the term with  onto its own side. This can be accomplished by subtracting the  to the right side of the equation. 

The result so far:

Then to finish getting  on its own, you divide  to the right side, ending up with:

So now looking at the question, we know that , so in order to figure out  we need to plug  into the equation.

This gives us .

So this gives us two possible answers:

So the derivative of a natural log  is always equal to  or one over whatever is inside the natural log. In this case  is inside the natural log, so the derivative of  should be:

But since the inside of the natural log is a function as well, this is the chain rule and the derivative of the natural log will be multiplied by the derivative of the inside, in this case , which is . 

So the final derivative is 

This problem involves using product rule, and chain rule.

By product rule,

Then, using power rule, and ten chain.

Another application of chain rule to get at the angle,

Taking the derivative of the angle and then simplifying, we get

The derivative of the function is equal to

and was found using the following rules:

, , , , 

Note that the chain rule is used for the exponential function (the secant is the inner function) and for the cosine function (the linear term is the inner function). 

The derivative of the function is equal to

and was found using the following rules:

, , , 

Note that the chain rule was used on the natural logarithm derivative and the derivative of the cosine function.

The first step in finding our second derivative is finding the first.

For our function  we note that it is a composite function, and therefore requires the use of the chain rule. The chain rule states if  then .

In our case this becomes:

To find the second derivative we must take the derivative of this function we've now computed. To do this we will require both the chain rule as stated, and the product rule since our function is of the form , our derivative will be of the form .

Therefore, our second derivative is:

or equivalently,

We start by taking the derivative of both sides of the equation, and proceeding as follows,

.