derivative of x^n

derivative of

x

n

Recall the typical

derivative

formula

is

d

⁢

f

d

⁢

x

=

lim

h

→

0

⁡

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

h

.

The derivative of

x

n

can be computed directly from this formula utilizing the

binomial theorem

, see for

instance

an alternative proof of the deriviative of

x

n

(

http://

planetmath

.org/AlternativeProofOfDerivativeOfXn

).

However, to avoid invoking the binomial theorem one can often make use of alternative definitions of the derivative which are justified by inspecting a diagrams and/or through the use of

algebra

. Instead of using

h

, use

h

=

x

-

a

so that

a

→

x

is the same as

h

→

0

. This gives the formula

d

⁢

f

d

⁢

x

=

lim

a

→

x

⁡

f

⁢

(

x

)

-

f

⁢

(

a

)

x

-

a

.

This is the standard slope formula between the two points

(

a

,

f

⁢

(

a

)

)

and

(

x

,

f

⁢

(

x

)

)

only now we let

a

approach

x

.

From this formula the casual rule

d

d

⁢

x

⁢

(

x

n

)

=

n

⁢

x

n

-

1

for

positive integer

values of

n

can be easily derived.

First notice that

(

x

-

a

)

⁢

(

x

n

-

1

+

x

n

-

2

⁢

a

+

⋯

+

x

⁢

a

n

-

2

+

a

n

-

1

)

=

x

n

-

a

n

.

Therefore

d

d

⁢

x

⁢

(

x

n

)

=

lim

a

→

x

⁡

x

n

-

a

n

x

-

a

=

lim

a

→

x

⁡

(

x

n

-

1

+

x

n

-

2

⁢

a

+

⋯

+

x

⁢

a

n

-

2

+

a

n

-

1

)

=

n

⁢

x

n

-

1

.

When

n

is not a positive integer the proof typically depends on

implicit differentiation

as follows:

y

=

x

n

;

ln

⁡

y

=

ln

⁡

x

n

=

n

⁢

ln

⁡

x

;

y

′

y

=

n

⁢

1

x

;

y

′

=

n

⁢

y

x

=

n

⁢

x

n

-

1

.

For the theoretically inclined this solution can be disappointing because it depends on a proof for the derivative of

ln

⁡

x

. Most texts simply redefine

ln

⁡

x

as the

integral

of

1

/

x

or in some

similar

fashion delay an honest proof.

For this reason it is often instructive to prove the

power rule

in stages depending on the type of

exponents

. Having proven the result for

n

a positive integer, one can extend this to

-

n

using the

product rule

.

To begin with observe

1

=

x

n

⁢

x

-

n

. Therefore

0

=

d

d

⁢

x

⁢

(

1

)

=

d

d

⁢

x

⁢

(

x

n

⁢

x

-

n

)

=

d

d

⁢

x

⁢

(

x

n

)

⁢

x

-

n

+

x

n

⁢

d

d

⁢

x

⁢

(

x

-

n

)

=

n

⁢

x

n

-

1

⁢

x

-

n

+

x

n

⁢

d

d

⁢

x

⁢

(

x

-

n

)

=

n

x

+

x

n

⁢

d

d

⁢

x

⁢

(

x

-

n

)

.

Now solve for

d

d

⁢

x

⁢

(

x

-

n

)

.

d

d

⁢

x

⁢

(

x

-

n

)

=

-

n

x

⁢

1

x

n

=

(

-

n

)

⁢

x

(

-

n

)

-

1

.

Likewise

fractional powers

can also be accommodated without the use of

ln

⁡

x

by beginning with the property: given

1

/

b

a

rational number

then

x

=

(

x

1

/

b

)

b

.

Using the

chain rule

we can prove the power rule for

x

1

/

b

as follows.

1

=

d

d

⁢

x

⁢

(

x

)

=

d

d

⁢

x

⁢

(

(

x

1

/

b

)

b

)

=

b

⁢

(

x

1

/

b

)

b

-

1

⁢

d

d

⁢

x

⁢

(

x

1

/

b

)

.

Once again solve for

d

d

⁢

x

⁢

(

x

1

/

b

)

d

d

⁢

x

⁢

(

x

1

/

b

)

=

1

b

⁢

(

x

1

/

b

)

b

-

1

=

1

b

⁢

x

1

/

b

⁢

x

-

1

=

1

b

⁢

x

1

/

b

-

1

.

Finally, the derivative of

x

a

/

b

for any

fraction

a

/

b

is done by observing that

x

a

/

b

=

(

x

a

)

1

/

b

so indeed the chain rule once again solves the problem.

Title

derivative of

x

n

Canonical name

DerivativeOfXn

Date of creation

2013-03-22 15:50:10

Last modified on

2013-03-22 15:50:10

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

16

Author

Algeboy (12884)

Entry type

Theorem

Classification

msc 26B05

Classification

msc 26A24

Synonym

Power rule

Related topic

DerivativesByPureAlgebra

Related topic

AlternativeProofOfDerivativeOfXn

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