Derivative of a Polynomial

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).

The derivative of xⁿ is n⋅x^{n + 1}.

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

This is true for any real number n
except for n = 0. (= a constant term)

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.

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.

Write the coefficient 2.

See x⁷.
Write the exponent 7.
And write x^{7 - 1} = x⁶.

Write the coefficient -5.

See x, which is x¹.
Write the exponent 1.
And write x^{1 - 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.

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 x^{3 - 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.

The exponent is e.
So write e.

Write x^{e - 1}.

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