Because we are differentiating a function within another function, we must use the chain rule, which states that

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Given the equation

,

we can deduce that

and

.

By plugging these into the chain rule, we conclude that

.

We proceed as follows.

. (Start)

. (Product rule)

. (The first derivative uses the Chain rule. The 2nd one uses the basic power rule.)

.

.

Although some factoring could be done at this point, we will not do so.

For a chain rule derivative, we need to work our way inward from the very outermost function. First, we need to do a power rule for the outer exponent. Then, we multiply that by the derivative of the inside.

For a chain rule derivative, we take the derivative of the outside function (leaving the inside function unchanged). Then, you multiply that by the derivative of the inside.

For a chain rule derivative, we need to take the derivative offirst. Then, we multiply by the derivative of the inside function. In other words, the general chain derivative of the function is:

The exponential function is the only derivative that always returns the original function. Therefore, we only need to multiply that by the derivative of the new exponent.

.

Here, we need to take the derivative of a square root function first. Then, we will multiply that by the derivative of the function under the square root.

Remember:by another chain rule.

Now, let's make use of our trigonometric identities:

Therefore, we can simplify our answer:

Note, you could have used this trigonometric identity from the very beginning, making the problem much easier!

For implicit differentiation, we need to take a derivative from left to right of our function. The only difference is that any time we take the derivative of afunction, we need to explicitly writeafter it. Then, we will use algebra to solve for that

Here, just move everything other than the term we want to the right. Then, divide by the coefficient.

For implicit differentiation, we need to take the derivative of every term, including numbers. Whenever we take a derivative of a function of, we need to multiply it by. Then, we will rearrange the equation to solve for

In the last step, I canceled all of the negative signs.

This is a pure problem on understanding how chain rules work for derivatives.

First thing we need to remember is that the derivative ofis.

When we are taking the derivative of, we can first pull out the 2 in the front and we treatas.

This way, the derivative will become,

which is.