calculus review – derivatives · you need to review the formulas commonly used in finding...

Calculus Review – Derivatives

© Arizona State University, Department of Mathematics and Statistics 1

Objectives: At the end of this lesson, you should be able to: 1. Find the derivative of functions involving sums, products, quotients and composition of polynomial,

exponential and logarithmic structures. 2. Evaluate the derivative (find the slope) at any point in the domain of the function.

Background

You need to review the formulas commonly used in finding derivatives. I know you don’t like the technical definition. It took me about a year to get it down really solid. However, there are times in business where you must use the definition, because the problem set is discrete. Here’s the formula again:

( )( ) ( )

0

' limh

f x h f xf x

h!

+ "=

Just as a reminder, it says 1. Calculate the functional value at a point in domain x and at a point h units distant from it. This creates

the change in output or rise. 2. Create the ratio of the result of step 1 and h. Now you have rise over run, or the average rate of

change across the h units of domain from x to (x + h). 3. If it is possible, shrink h to as close to zero as possible. That creates the functions limit at point x.

4. If all the math works, we have the derivative, also known as the instantaneous rate of change, the slope of the tangent line, or the rate of change at the point x, and maybe another three or four common names.

Let’s get blunt. If this is a mystery to you, you need to get help now! From this powerful calculation we developed a host of the following rules:

The Power Rule ( ) 1n ndx n x

dx

!= " .

Where ( ) ( ),u f x v g x= = are both continuous functions, we get these results:

1. Product Rule ( ) ' 'du v u v u v

dx! = ! + !

2. Quotient Rule 2

' 'd u u v u v

dx v v

! " !# $=% &

' (

3. Chain Rule .dy dy du

dx du dx= and the Generalized Power Rule ( ) 1

'n ndu n u u

dx

!= " "



We created special cases to simplify our work. They all come out of the Chain Rule, which may explain why I’ve never really memorized them!

1. Exponential Rules: ( )x xde e

dx= and ( ) '

u ude e u

dx= ! , ( ) ln

x xda a a

dx= ! ,

( ) ln 'u uda a a u

dx= ! !

2. Logarithmic Rules: ( )1

lnd

xdx x

= and ( )'

lnd u

udx u

=

Finally, if ( )y f x= , the derivative can be written as ( )( )

' 'df xdy

y f xdx dx

= = = .

Discussion We developed the process of finding the derivatives of non-linear functions because the slope of a curve changes at each point on the graph.

As we said, the formula for finding the first derivative is ( )( ) ( )

0

' limh

f x h f xf x

h!

+ "= . Remember how it

was developed from the original formula 2 1

2 1

y ym

x x

!=

! for finding the slope of a line joining two points

( )1 1,x y and ( )2 2

,x y . All other formulas we developed later were based on this.

The only problem with the formula is that it has too many names !The derivative is defined to be the slope of the tangent line at any point on the graph so long as it is a unique value.* It is also considered as the rate of change of the variable y with respect to the variable x .

* Recall that the simple function f (x) = x is beautifully continuous. However, it does not have a derivative at zero, because the slope of the tangent line has no unique value there.

Example: Suppose we have the function, ( ) 22 5 4f x x x= ! + .

From algebra, you know this is a convex quadratic function that is that opens up. The slope is negative on the left of the vertex and positive on the right of the vertex. However, the slope is different at every point on the graph. If we use the definition of the derivative above, we can find the appropriate formula that defines the slope at any point on the graph.

( )( ) ( ) ( ) ( ) ( )

2 2

0 0

2 5 4 2 5 4' lim lim

h h

x h x h x xf x h f xf x

h h! !

+ " + + " " ++ "= =

We can now simplify using basic algebra.

( )2 2 2

0

2 2 2 2 5 5 4 2 5 4' lim

h

x xh h x h x xf x

h!

+ " + # # + # + #=



( )( )

( )2

0 0 0

4 2 54 2 5' lim lim lim 4 2 5 4 5

h h h

h x hxh h hf x x h x

h h! ! !

+ "+ "= = = + " = "

Hence, ( )' 4 5f x x= ! is the formula that defines the derivative of the function ( ) 22 5 4f x x x= ! + .

Let’s apply the rule at the point ( )2,22! is on the graph.

1. Make sure the point is on the graph! Since ( ) ( ) ( )2

2 2 2 5 2 4 8 10 4 22f ! = ! ! ! + = + + = , the point is on the graph.

2. Now, evaluate the derivative. At this point the derivative is ( ) ( )' 2 4 2 5 8 5 13f ! = ! ! = ! ! = ! . We

can say that the rate of change of y with respect to x at ( )2,22! is -13.

As much as I love to do it, this limit process ( )( ) ( )

0

' limh

f x h f xf x

h!

+ "= becomes tedious, dangerously

fraught with calculation error, and time-consuming for anything beyond a quadratic. We cannot even use it directly for most exponential and logarithmic functions! We developed all the other formulas to save time and reduce using more complex algebra.

Examples Let’s look at some (many) examples. The only trick is to use the appropriate formulas.

1. ( ) 250f x x=

The definition would be tedious to use. The power rule ( ) 1n ndx n x

dx

!= " allows us to find the

derivative more quickly. It is ( )250 249250

dx x

dx= .

2. ( ) 5 34 2 7 9f x x x x= ! + !

We can use the power rule on each of the 4 terms in this function.

Remember that ( ) 5 34 2 7 9f x x x x= ! + ! can be seen as the sum of four power functions.

So, ( ) 4 2 0 1 4 2' 4 5 2 3 7 1 0 9 20 6 7f x x x x x x x!

= " ! " + " ! " = ! + as we take the derivatives term-by-term.

3. ( ) ( )( )2 33 4 8 2 5f x x x x= + ! !

This can be written as ( )f x u v= ! where 23 4 8u x x= + ! and 3

2 5v x= ! .

The formula that fits the function is the product rule ( ) ' 'du v u v u v

dx! = ! + ! .

We can now find the derivative of each of these two new and simpler functions in the same way as before and then substituting them into the formula to get the derivative ( )'f x .



Let ' 6 4u x= + and 2' 6v x=

Now using the product rule,

( ) ( ) ( )( ) ( )( )3 2 2' ' ' 6 4 2 5 3 4 8 6

df x u v u v u v x x x x x

dx= ! = ! + ! = + " + + " . Simplify using basic

algebra.

4. ( )2

2

6 5 1

3 5

x xf x

x

! +=

!

Write it as ( )u

f xv

= where 26 5 1u x x= ! + and 2

3 5v x= ! .

The formula that now fits is the quotient rule 2

' 'd u u v u v

dx v v

! " !# $=% &

' (.

Find ' 12 5u x= ! and ' 6v x= .

The quotient rule gives

( )( )( ) ( )( )

( )

2 2

222

12 5 3 5 6 5 1 6' ''

3 5

x x x x xd u u v u vf x

dx v v x

! ! ! ! +" ! "# $= = =% &

' ( ! which can be

simplified using basic algebra.

5. Suppose 125 4y u= + and 3

2 1u x= ! .

The derivative of the composition ( ) ( )12

35 2 1 4y f x x= = ! + is difficult to find.

Use the chain rule: .dy dy du

dx du dx=

We get 11

' 60dy

y udu

= = with 2

' 6du

u xdx

= =

Applying them, f ' x( ) =

dy

dx=

dy

du.du

dx= 60u11

!6x2= 60 2x3

"1( )11

!6x2

Simplifying leads to f ' x( ) = 360x2

2x3!1( )

11

.

If you don’t simplify, be in for a stunning problem when you go for the second or third derivative!



6. ( ) ( )8

22 1y f x x x= = + !

Look at it as ( ) ( )8

2 82 1y f x x x u= = + ! = where 2

2 1u x x= + ! .

Now, ( ) 1'

n ndu n u u

dx

!= " " (one instance of the chain rule) fits perfectly with ' 2 2u x= +

That gets us ( ) ( ) ( )7

2' ' 8 2 1 2 2y f x x x x= = + ! " +

7. If 25 2 1

xy x e= ! + then ' 5 2 2 10 2

x xy x e x e= ! " = "

8. Let 2

2 7x xy e

+= .

This can be seen as u

y e= where 2

2 7u x x= + .

Hence, ' 2 2 7 4 7u x x= ! + = + . So, ( )2

2 7' 4 7

x xy e x

+= ! +

by the chain rule.

9. Look at y = 9x . It fits d

dxau( ) = au ! ln a( ) "u and !y = 9

x" ln 9 .

10. 2

4 37x x

y!

= fits ( ) ln '

u uda a a u

dx= ! !

So, ( )2

4 3' 7 ln 7 8 3

x xy x

!= " " ! by the chain rule.

11. 24 ln 6 2

xy x x e= ! + leads to

1 4' 4 6 2 2 12 2

x xy x e x e

x x

! "= # $ # + = $ +% &

' (

12. ( )2ln 6 5 11y x x= + ! fits ( )

'ln

d uu

dx u= where 2

6 5 11u x x= + ! . Hence ' 12 5u x= +

and 2

12 5'6 5 11

xy

x x

+=

+ !.

Functions can be constructed where one or more of these rules do fit simultaneously. You need to make a conscious effort to recognize the one or more of these rules that fit. This is necessary to be able to find the derivative of complex functions. If you can use these rules appropriately, you will have a solid background when finding derivatives of functions having two or more independent variables. You should practice finding derivatives of a few functions before going on.

calculus review – derivatives · you need to review the formulas commonly used in finding...

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