The Concept of Derivative - Rules of
Differentiation
Hong Jengei, Handong University
October 10, 2018
Hong Jengei, Handong University The Concept of Derivative - Rules of Differentiation
Rules of Differentiation for a Function of One Variable
Constant Function Rule : The derivative of a constant function is
identically zero.
dy df (x)
If y = f (x) = k, = = f 0 (x) = 0.
dx dx
Hong Jengei, Handong University The Concept of Derivative - Rules of Differentiation
Rules of Differentiation for a Function of One Variable
Power Function Rule : The derivative of a power function y = x n is
nx n−1 .
dy df (x)
If y = f (x) = x n , = = f 0 (x) = nx n−1 .
dx dx
Hong Jengei, Handong University The Concept of Derivative - Rules of Differentiation
Rules of Differentiation for a Function of One Variable
Generalized Power Function Rule : When a multiplicative constant c
appears in the power function, so that y = f (x) = cx n , its
derivative is
dy
= cnx n−1 .
dx
Hong Jengei, Handong University The Concept of Derivative - Rules of Differentiation
Practice
Solve for the derivatives of the followings
y = 5x 0
y = 2x 3
y = −2x −3
1
y=
x3
√
y= x
Hong Jengei, Handong University The Concept of Derivative - Rules of Differentiation
Rules of Differentiation Involving Two or More Functions
of The Same Variable
Sum-Difference Rule : The derivative of a sum(difference) of two
function is the sum(difference) of the derivative of the two functions:
d(f (x) ± g (x)) df (x) dg (x)
= ±
dx dx dx
Hong Jengei, Handong University The Concept of Derivative - Rules of Differentiation
Practice
Solve for the derivatives of the followings
y = 2x 3 + 13x 2 − x
y = 7x 4 + 2x 3 − 3x + 37
Hong Jengei, Handong University The Concept of Derivative - Rules of Differentiation
Rules of Differentiation Involving Two or More Functions
of The Same Variable
Product Rule : The derivative of a product of two function is equal
to the first function times the derivative of the second function plus
the second function times the derivative of the first function.
d(f (x)g (x))
= f (x)g 0 (x) + f 0 (x)g (x)
dx
Hong Jengei, Handong University The Concept of Derivative - Rules of Differentiation
Practice
Solve for the derivatives of the followings
y = (2x + 3)3x 2
Hong Jengei, Handong University The Concept of Derivative - Rules of Differentiation
Rules of Differentiation Involving Two or More Functions
of The Same Variable
f (x)
Quotient Rule : The derivative of the quotient of two function g (x) is
d( gf (x)
(x) ) f 0 (x)g (x) − f (x)g 0 (x)
=
dx g 2 (x)
Hong Jengei, Handong University The Concept of Derivative - Rules of Differentiation
Practice
Solve for the derivatives of the followings
2x − 3
y=
x +1
5x
y=
x2 + 1
ax 2 + b
y=
cx
Hong Jengei, Handong University The Concept of Derivative - Rules of Differentiation
Rules of Differentiation Involving Functions of Different
Variables
Chain Rule : If we have a function z = f (y ), where y is in turn a
function of another variable x, say, y = g (x), then the derivative of
z with respect to x is equal to the derivative of z with respect to y
times the derivative of y with respect to x.
dz dz dy
= = f 0 (y )g 0 (x)
dx dy dx
Hong Jengei, Handong University The Concept of Derivative - Rules of Differentiation
Practice
dz
Solve for dx
z = 3y 2
y = 2x + 5
dz
Solve for dx
z =y −3
y = x3
Hong Jengei, Handong University The Concept of Derivative - Rules of Differentiation
Rules of Differentiation for a Function of One Variable
Inverse function Rule : If the function y = f (x) represents a
one-to-one mapping, the function f will have an inverse function
x = f −1 (y ). Then,
dx 1
= dy
dy dx
Hong Jengei, Handong University The Concept of Derivative - Rules of Differentiation
Practice
dx
Solve for dy
y = 5x 4 + 1
Hong Jengei, Handong University The Concept of Derivative - Rules of Differentiation
