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) = limh → 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 xn: Power Rule

The derivative of xn is n⋅xn + 1.

First write n.
Then decrease the exponent n to (n - 1): xn → xn - 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 x2.

So 3x2 is the derivative of 32.

Example 2

Example

Solution

Write the coefficient 2.

See x7.
Write the exponent 7.
And write x7 - 1 = x6.

Write the coefficient -5.

See x, which is x1.
Write the exponent 1.
And write x1 - 1 = x0.

+3 doesn't have a variable.
So +3 is a constant.

So the derivative of +3 is +0.

So the derivative of 2x7 - 5x + 3 is
2⋅7⋅x6 - 5⋅1⋅x0 + 0.

2⋅7⋅x6 = 14x6
-5⋅1⋅x0 = -5

So f'(x) = 14x6 - 5 is the derivative of f(x).

The derivative of -5x became -5.
From this, you can see that
the derivative of x (= x1) is 1.

Example 3

Example

Solution

Split the fraction
by dividing each term in the numerator by the denominator x3.

6x6/x3 = 6x3
-3x3/x3 = -3
+2x2/x3 = -2x-1
-1/x3 = -x-3

Quotient of Powers

Negative Exponent

Find y' by finding the derivative of each term.

Write the coefficient 6.

See x3.
Write the exponent 3.
And write x3 - 1 = x2.

-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⋅x2 - 0 + 2⋅(-1)⋅x-2 - (-3)⋅x-4.

6⋅3⋅x2 = 18x2
+2⋅(-1)⋅x-2 = -2/x2
-(-3)⋅x-4 = +3/x4

So y' = 18x2 - 2/x2 + 3/x4 is the derivative of y.

Example 4

Example

Solution

The exponent is e.
So write e.

Write xe - 1.

y' = exe - 1 is the derivative of y = xe.