DERIVATIVES USING THE DEFINITION

DERIVATIVES USING THE DEFINITION

Doing derivatives can be daunting at times, however, they all follow a general rule and can be

pretty easy to get the hang of. Let��s try an example:

Find the derivative of ! ! = ! ! , and then find what the derivative is as x approaches 0.

The first thing we must do is identify the definition of derivative. The definition is ! ! ! =

lim!��!

! ! !!(!)

!!!

. In our example, ! ! = ! ! . (Remember: we can find what !(!) is by

replacing every x in our equation with c). Now that we have our definition, let��s find the

derivative. Note: we must first simplify the equation down as far as possible before we take the

limit, or else it may skew our answer.

First we must replace ! ! with our function:

! ! ?! !

!! ? !!

= lim

!��!

!��! ! ? !

!?!

lim

Now we need to reduce our equation as much as possible. Since we are taking the limit of the

function, we are allowed to do this: ?

= lim

!��!

!+! !?!

?

!?!

= lim ! + !

!��!

Now we take the limit of the function at c: ?

= ! + ! = 2!

Now we are asked to find what derivative is as x approaches 0. For this, we need to now replace

every c in our solution with 0.

2! = 2 0 = 0

Therefore, our derivative is 2! and equals 0 as x approaches 0.

Note: there is another accepted definition of derivative which is ! ! ! = lim!��!

! !!! !!(!)

!

we let ! ! = ! ! , then:

! ! ! = lim

!��!

! + ? ! ? !!

?

?

! ! + 2!? + ?! ? ! !

?

!��!

?

= lim

2!? + ?!

= lim

?

!��!

?

= lim 2! + ? = 2!

!��!

We would then replace x with 0 to find our derivative as x approaches 0.

2! = 2 0 = 0

Notice, both definitions gave us the same solution.

For a video of this, please reference

. If

DERIVATIVES USING THE POWER RULE

Sometimes using the definition of derivative can be quite cumbersome, but luckily there is a

shortcut we can use to find the derivative. Let��s look at an example:

Determine the derivative of ! ! = ! ! + 6! ? 2 using the power rule.

The first thing we must do is remember what our power rule is:

!" ?! ! = ! ! ,

!?!" ?! ! ! = ! ? ! !!! ?

We must remember that the derivative of f(x) plus g(x) is the same as the derivative of f(x) plus

!

the derivative of g(x). That is !" ! ! + ! !

=

!" !

!"

+

!" !

!"

. So for our example:

!" !

! !

=

! + 6! ? 1 ?

!"

!"

=

! !!

! 6!

! 1

+

?

!"

!"

!"

Now we can take the derivative of each individual piece using the power rule:

?

?

?

! !!

!"

! !!

!"

! !

!"

= 2 ? ! !!! = 2!

= 1 ? 6 ? ! !!! = 6! ! = 6 *Note: !! is 1.

= 0 *Note: the derivative of all constants is zero.

Now we can solve our problem:

! ! ! = 2! + 6 + 0 ?

! ! = 2! + 6

For a video on this, please reference

DERIVATIVES USING THE PRODUCT RULE

Sometimes using the definition for finding a derivative can be cumbersome, especially when

there are multiple functions of x combined together. However, we have the product rule to help

us out. Let��s look at an example:

Determine the derivative of ! ! = !! ! .

The first thing we must do is remember the product rule:

!

!" = !! ! + !"��

!"

Now we must define what u and v are and find their derivatives (we are allowed to use the power

rule here):

! = !,

!! = 1 ?

! = !!,

!! = ! !

Now all we have to do is replace our values and simplify:

!

!! ! = !! ! + 1! ! ?

!"

= ! ! (! + 1)

*Note: This can be expanded to include more than two variables, i.e.

!

!"# = !"! ! + !! ! ! + !��!"

!"

For a video on this, please reference

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download