# Power Rule in Differentiation (Part 1)

How to solve the power rule in differentiation problems (when the exponent is a natural number): formulas, proofs, examples, and their solutions.

## Formulas

The derivative of *x*^{n} is *nx*^{n - 1}.

This is true when *n* is a natural number.

(Actually, this is true when *n* is a real number.

But you'll see the other cases later.)

Power rule in differentiation (Part 2)

Power rule in differentiation (Part 3)

Power rule in differentiation (Part 4)

The derivative of a constant number *C* is 0.

This is the case when *n* = 0.

(= the constant has no variable *x*.)

## Proofs

Let's see the proof of [*x*^{n}]' = *nx*^{n - 1}.

Use the definition of a derivative function.

Expand (*x* + *h*)^{n}.

Binomial theorem

Simplify the first two terms

using _{n}C_{0} = 1 and _{n}C_{1} = *n*.

And leave the other terms.

Combinations (_{n}C_{r})

Cancel the dark gray terms.

Divide each term in the numerator by *h*.

*nx*^{n - 1} term doesn't have *h*.

So, as *h* → 0, *nx*^{n - 1} remains.

The other terms all have *h*.

So, as *h* → 0, they all go to 0.

So [*x**n*]' = *nx*^{n - 1}.

Let's see the proof of [*C*]' = 0.

Use the definition of the derivative function.*f*(*x* + *h*) = *f*(*x*) = *C*.

The numerator is real 0.

So the limit is 0.

So [*C*]' = 0.

## Example 1

The exponent of *x*^{2} is 2.

So *f*'(*x*) is,

write the exponent, 2,

write *x*,

and reduce the exponent to, 2 - 1, 1.

So *f*'(*x*) = 2*x*.

## Example 2

The exponent of *x*^{5} is 5.

So *f*'(*x*) is,

write the exponent, 5,

write *x*,

and reduce the exponent to, 5 - 1, 4.

So *f*'(*x*) = 5*x*^{4}.

## Example 3

The exponent of *x* is 1.

So *f*'(*x*) is,

write the exponent, 1,

write *x*,

and reduce the exponent to, 1 - 1, 0.

So *f*'(*x*) = 1.

Zero exponent

## Example 4

*f*(*x*) is a constant function:

it has no variable *x*.

So *f*'(*x*) = 0.