By the Power Rule of derivatives, for any equation , the derivative .

With our function , where , we can therefore conclude that:

In order to find the derivative we will need to use the power rule on each term. The power rule states,

.

Applying this rule we get the following.

For any function , the first derivative .

Therefore, taking each term of :

To find the derivative of this function, we simply need to use the Power Rule. The Power rule states that for each term, we simply multiply the coefficient by the power to find the new coefficient. We then decrease the power by one to obtain the degree of the new term.

For example, with our first term, , we would multiply the coefficient by the power to obtain the new coefficient of . We then decrease the power by one from 4 to 3 for the new degree. Therefore, our new term is . We then simply repeat the process with the remaining terms.

Note that with the second to last term, our degree is 1. Therefore, multiplying the coefficient by the power gives us the same coefficient of 8. When the degree decreases by one, we have a degree of 0, which simply becomes 1, making the entire term simply 8.

With our final term, we technically have

Therefore, multiplying our coefficient by our power of 0 makes the whole term 0 and thus negligible.

Our final derivative then is

To find the derviative of this equation recall the power rule that states: Multiply the exponent in front of the constant and then subtract one from the exponent.

We can work individually with each term:

Derivative of

is,

For the next term:

Derivative of :

So answer is:

Anything to a power of 0 is 1.

For the next term:

Derivative of :

Any derivative of a constant is .

So the first derivative of

is

For any function , the first derivative .

Therefore, taking each term of :

To find the derivative of this function, use the power rule and the chain rule.

The power rule is:

The chain rule is to take the derivative of the inner function.

Apply the power rule for the function and the chain rule. The derivative of is .

The answer is:

To solve, you must use the product rule as outline below.

Product rule:

Thus,

Distribute the 2x and 2.

Combine like terms to simplify.

By the Power Rule of derivatives, for any equation , the derivative .

Given our function , where , we can conclude that

Because the original function is the quotient of 2 separate functions, we can use the Quotient Rule.

Quotient Rule:

If

,

then

.

gives us

, and .

Now

Or

.

Simplifying yields

or

.

We can factor out to see that finally

.

Because the two terms and are different, no more terms can cancel.