To find the derivative of the function, we use the quotient rule, which is 

Applying this to the function from the problem statement, we get

To find the derivative of the function, we use the product rule, which by definition is

In this case, we can split up the product such that

This question requires us to understand two things.

First, the derivative of  is always , such that the exponent contains a single variable (any other operation or numerical factors could cause the chain rule to come into play; a later topic)

Second, we must understand the product rule for derivatives. The product rule works as follows:

Understanding these two concepts allows us to tackle the derivative of the given function. 

This simplifies to:

We are finished taking the derivative of the product!

There are two steps to this problem:

1) Take the first derivative of the function:

 (applying the quotient rule)

 (simplifying and combining terms)

Thus, the first derivative is as follows:

2) Take the second derivative of the original function (or the first derivative of the first derivative we just found)

 (applying the quotient rule)

 (simplifying and combining terms)

Thus, our second derivative is as follows:

So the derivatives of the original function "r" should be: 

We are given the function:

Applying the general power rule, we can take the derivative

 (the 3e term is removed because "e" is a constant, and deriving a constant gives us zero!)

 (further simplifying)

Thus, the derivative of our original function, and the correct answer, is:

We are given the function:

The trick to making this function more approachable is to rewrite the function. The negative 1 on the right-most term allows us to rewrite the function as:

We can now apply the quotient rule for derivatives, and derive the function:

This derivative is quite messy, but we can simplify the numerator further!

Simplifying once more, we get our answer:

We are given the function:

As the function given to us is a quotient, we must use the quotient rule for derivatives to get the derivative of the function.

Technically, this derivative is correct, but it is not at all optimal. Let's clean things up and simplify our answer further:

Thus, we combine our terms, do our last simplification, and reach our answer:

To answer this question, we must understand two key concepts:

First, we must understand that the derivative of  is , as long as the exponent is a variable by itself (no coefficients or other operations). Now, x can be replaced with any variable letter, really. So, the derivative of  is just  .

Second, we must understand the quotient rule of derivation. The quotient rule is very important to calculus and it looks like this:

This rule shows you how to take the derivative of a quotient, hence the quotient rule.

(Here's a quick rhyme to remember this rule, if that helps:

"Low "dee" high less high "dee" low,   ["dee" meaning derivative of]

draw the line and down below,

the denominator squared will surely go!" )

Combining the derivative of "e" raised to a variable, and the quotient rule, we can carry out our solution:

   (quotient rule)

(you can leave the answer like this, or simplify further to the next line)

And you are done!

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If you understand the first two derivatives, as well as number three and number four - the quotient and product rules for derivatives, respectively - this problem is not that difficult! However, it is easy to make a mistake keeping track of the term grouping (parentheses, brackets, and such). You can do it!

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Given:

Set the problem up as a quotient rule:

 (Required knowledge)

Thus:

We run into a small problem with the  in the above line. To find the derivative of this term, we must use the product rule (or know the derivative from memory)

We can now continue the problem, plugging in the above portion for the d(sin^2(x))

 break the terms up

Thus, our final, simplified answer should be:

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                                          CORRECT ANSWER

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                                              PROBLEMS?

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If you had problems with the concepts in this question, refer to the "required knowledge" at the top of the explanation for this problem. If you have any other questions on the algebra portion, or need clarification on anything done in this explanation, ask your Varsity Tutor for assistance or further explanation!

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Given: 

Thus, our answer is:

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                                        CORRECT ANSWER

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