Find the First Derivative of a Function - Pre-Calculus

For this problem we need to use the Chain Rule.

The Chain Rule states to work from the outside in. In this case the outside function is and the inside function is . The derivative then becomes the outside function times the derivative of the inside function.

Thus, we use the following formula

.

In our case , and .

Therefore the result is,

.

Note that is defined for all

Using Logarithm Laws we can write as .

Using the Chain Rule, and the Product Rule . We have:

.

Note that,

.

Now replace and we get our final answer:

In order to take the first derivative of the polynomial, all we need to know is how to apply the power rule to a simple term with an exponent:

The formula above tells us that to take the derivative of a term with coefficient and exponent , we simply multiply the term by and subtract 1 from in the exponent. With this in mind, we'll take the derivative of the function in the problem term by term:

We can see that our function involves a series of terms raised to a power, so we will need to apply the power rule as well as the chain rule to find the derivative of the function. First we apply the power rule to the terms in parentheses as a whole, and then we apply the chain rule by multiplying that entire result by the derivative of the expression in parentheses alone:

For the function

The first derivative is

So for

the first derivative is

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

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

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

Given our function , where , we can conclude that

To take the derivative of a polynomial, we'll need to apply the power rule to a term with a coeffient and an exponent :

Applying this rule to each term in the polynomial:

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

Recall the power rule that states to multiply by the exponent in front of the constant and then subtract the exponent by 1.

So lets take the derivative of this in sections:

Derivative of :

Result is:

Lets take the derivative of the next term:

Derivative of :

The power of something to is .

Our result is:

Recall the quotient rule:

So for:

Find their derivatives:

Plug in to the formula:

The derivative of a constant is .

Since we take the derivative with respect to we only apply power rule to terms that contain . Since all the terms contain , we treat as a constant.

The derivative of a constant is always .

So the derivative of

is zero.

Apply Power Rule.

First Derivative of :

Result is

So the first derivative of is

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 derivative of this function, we will have to apply both the power rule and the chain rule. First we apply the power rule, bringing the exponent of the entire term to the front as a coefficient and subtracting 1 from the new exponent. Then we apply the chain rule, multiplying that entire term by the derivative of just the expression inside the parentheses:

To take the derivative of this function, we can simply apply the power rule to most of the terms. For the first two, we apply the power rule, multiplying the coefficient out front by the exponent of the term, and then subtracting 1 from the new exponent. When we get to ln(x), we must remember that the derivative of this term is 1/x. Finally, we have 18, which is just a constant, and the derivative of a constant is always 0:

For the first term we can simply apply the power rule, multiplying the coefficient by the exponent of the term, which gives us 1 as the new coefficient, and subtracting one from the new exponent. When we get to our exponential terms, we have to remember to apply the chain rule to each one. The derivative of is just , because when we apply the chain rule, we multiply the term by the derivative of its exponent, which for x is just 1. For our next term, the exponent is 2x, so its derivative is 2, which we multiply by the whole term to give us . Finally our last term has an exponent of , which has a derivative of , so we multiply this by the whole term to give us :

For any function , the first derivative .

Therefore, taking each term of :