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.