# Derivative of a Polynomial

How to find the derivative of a polynomial function: definition, formula (power rule), 4 examples, and their solutions.

## Derivative Function

### Definition

f'(x) is the derivative function of f(x).

So f'(x) = lim_{h → 0} [f(x + h) - f(x)]/h.

Derivative - Definition

The derivative function is written as

f'(x), y', dy/dx, (d/dx)f(x).

## Formula

### Derivative of x^{n}: Power Rule

The derivative of x^{n} is n⋅x^{n + 1}.

First write n.

Then decrease the exponent n to (n - 1): x^{n} → x^{n - 1}.

This is true for any real number n

except for n = 0. (= a constant term)

### Derivative of a Constant

The derivative of a constant is 0.

This is true because

y = constant is a horizontal line.

So its slope is 0.

So its derivative is 0.

## Example 1

### Example

### Solution

The exponent is 3.

So write 3.

Decrease the exponent 3 to, 3 - 1, 2.

So write x^{2}.

So 3x^{2} is the derivative of 3^{2}.

## Example 2

### Example

### Solution

Write the coefficient 2.

See x^{7}.

Write the exponent 7.

And write x^{7 - 1} = x^{6}.

Write the coefficient -5.

See x, which is x^{1}.

Write the exponent 1.

And write x^{1 - 1} = x^{0}.

+3 doesn't have a variable.

So +3 is a constant.

So the derivative of +3 is +0.

So the derivative of 2x^{7} - 5x + 3 is

2⋅7⋅x^{6} - 5⋅1⋅x^{0} + 0.

2⋅7⋅x^{6} = 14x^{6}

-5⋅1⋅x^{0} = -5

So f'(x) = 14x^{6} - 5 is the derivative of f(x).

The derivative of -5x became -5.

From this, you can see that

the derivative of x (= x^{1}) is 1.

## Example 3

### Example

### Solution

Split the fraction

by dividing each term in the numerator by the denominator x^{3}.

6x^{6}/x^{3} = 6x^{3}

-3x^{3}/x^{3} = -3

+2x^{2}/x^{3} = -2x^{-1}

-1/x^{3} = -x^{-3}

Quotient of Powers

Negative Exponent

Find y' by finding the derivative of each term.

Write the coefficient 6.

See x^{3}.

Write the exponent 3.

And write x^{3 - 1} = x^{2}.

-3 is a constant.

So its derivative is 0.

Write the coefficient +2.

See x^{-1}.

Write the exponent (-1).

And write x^{-1 - 1} = x^{-2}.

Write the coefficient -.

See x^{-3}.

Write the exponent (-3).

And write x^{-3 - 1} = x^{-4}.

So y' = 6⋅3⋅x^{2} - 0 + 2⋅(-1)⋅x^{-2} - (-3)⋅x^{-4}.

6⋅3⋅x^{2} = 18x^{2}

+2⋅(-1)⋅x^{-2} = -2/x^{2}

-(-3)⋅x^{-4} = +3/x^{4}

So y' = 18x^{2} - 2/x^{2} + 3/x^{4} is the derivative of y.

## Example 4

### Example

### Solution

The exponent is e.

So write e.

Write x^{e - 1}.

y' = ex^{e - 1} is the derivative of y = x^{e}.