First step is to figure out what  is. This is done by inputting  wherever there is an  in , which gives you 

.

Then you start on the outermost function, the sine function, and work your way inwards.

Again, chain rule for  is:

,

which gives us for this problem

So implicit differentiation means that you take the derivative of every variable in the same way, with its respective rate of change, dx or dy in this case, coming out multiplied on the end.

You take the derivative of the whole equation and the result is:

The first step to solve for dy/dx is to create a dy/dx term, which means that you need to divide the entire equation by dx, to get it on the bottom. Each term becomes divided by dx, so the dx cancels with itself and you are left with two terms that have . The result is:

At this point, you want to get  on its own side, so this involves subtracting the two terms that aren't multiplied by  onto the right side of the equation.

After that, you factor out the  and end up with:

Then, the last step is to divide over the two term block to the other side to get  on its own.

The chain rule of the derivative always deals with the composition of two or more functions.

In this case we can identify two, 

and 

.

So  is the composition of these two such that: 

With chain rule, you always start on the outermost function and work your way inward, which in this case is: 

Always the derivative of the outermost evaluated at the inner, multiplied by the derivative of the inner.

We are going to use three rules along with the chain rule:

So then, using our first rule and the chain rule

then using our second rule and chain rule

then using our third rule (no chain rule this time)

Then we rearrange the equation for simplification,

We are going to use two rule and the chain rule

Then, using rule one and rule two (don't forget the chain rule)

Then we simplify

We will use the chain rule combined with our power rule:

Then by using this rule

Then applying the power rule to each element in the second parenthesis, 

We will be using the chain rule along with the following rules:

Looking at , first we will apply rule 3 with the chain rule,

Now we apply rule 1 (with chain rule) and rule 3 (no chain rule needed here),

Lastly we apply rule 2 (no chain rule needed),

Then we simplify as much as we can, and of course, what we have just calculated is the derivative of the original,

We will be using the following rule and the chain rule:

Also remember that

Now, looking at our function we are to find the derivative of,

We must use our first rule and the chain rule,

Let's rewrite that a little using our second rule,

Now let's use our first rule again (no further chain rule is required)

And now we just need to simplify,

So

We are going to use the following rules and the chain rule:

Now, looking at our original function,

can be written as

Then

Now we finish up by using the second rule backwards and then the first rule on the remainder (no more chain rule is required),

And finally, we have the derivative

Compute the derivative of the following function:

To find this derivative, we are going to use the chain rule:

What this means, is that we will take the derivative of the "outside" *in this case the  bit and keep the "inside" the same. Then we multiply all of this by the derivative of the inside.

So, let's jump in.

We can easily do our f' part. Doing so yields

Next, calculate the derivative of the inside.

Put that back together with the above part to get:

Clean it up a bit to get.

Which we will leave as...