Derivative of a Polynomial

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.

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

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.