Class notes Differential Calculus

Module 5: HIGHER ORDER DERIVATIVES AND IMPLICIT DIFFERENTIATION

5.1 Higher Order Derivatives

𝑑𝑦
If the function 𝑦 = 𝑓(𝑥) is differentiated, then its derivative is sometimes called the first derivative
𝑑𝑥
𝑑𝑦 𝑑𝑦
of y with respect to x. If the function is differentiable then the derivative of is called the second
𝑑𝑥 𝑑𝑥
𝑑2𝑦
derivative of y with respect to x denoted by or 𝑦 ′′ . Similarly, the third derivative of 𝑦 = 𝑓(𝑥) is defined as
𝑑𝑥 2
𝑑2𝑦 𝑑3𝑦
the derivative of 2
with respect to x, denoted by , and so on.
𝑑𝑥 𝑑𝑥 3

The nth derivative of the function 𝑦 = 𝑓(𝑥) where n is a positive integer, is the derivative of the (n-1)th
derivative of 𝑦 = 𝑓(𝑥). We denote the nth derivative of 𝑦 = 𝑓(𝑥) by 𝑦 (𝑛) .


Some notations for higher derivatives are the following:
𝑑𝑦 𝑑
first derivative: , 𝑦 ′ , 𝑓 ′ (𝑥), 𝑓(𝑥), 𝐷𝑥 𝑓(𝑥)
𝑑𝑥 𝑑𝑥
𝑑2 𝑦 𝑑2
second derivative: , 𝑦 " , 𝑓 " (𝑥), 𝑓(𝑥), 𝐷2 𝑥 𝑓(𝑥)
𝑑𝑥 2 𝑑𝑥 2




𝑑𝑛 𝑦 𝑑𝑛
nth derivative: , 𝑦 (𝑛) , 𝑓 𝑛 (𝑥), 𝑓(𝑥), 𝐷𝑛 𝑥 𝑓(𝑥)
𝑑𝑥 𝑛 𝑑𝑥 𝑛


Example 1 Find all the derivatives of the function 𝑦 = 5𝑥 6 − 4𝑥 5 + 3𝑥 4 − 10𝑥 3 + 12𝑥 − 6

Solution:+
Given that 𝑦 = 5𝑥 6 − 4𝑥 5 + 3𝑥 4 − 10𝑥 3 + 12𝑥 − 6
𝑦 ′ = 30𝑥 5 − 20𝑥 4 + 12𝑥 3 − 30𝑥 2 + 12
𝑦 ′′ = 150𝑥 4 − 80𝑥 3 + 36𝑥 2 − 60𝑥
𝑦 ′′ ′ = 600𝑥 3 − 240𝑥 2 + 72𝑥 − 60
𝑦 (4) = 1800𝑥 2 − 480𝑥 + 72
𝑦 (5) = 3600𝑥 − 480
𝑦 (6) = 3600

𝑑2𝑦 𝑥 2 −𝑥+2
Example 2 Find 2
given that 𝑦 =
𝑑𝑥 𝑥 2 +1
Solution:
𝑥 2 −𝑥+2
𝑦=
𝑥 2 +1
𝑑𝑦 (𝑥 2 +1)(2𝑥−1)−(𝑥 2 −𝑥+2)(2𝑥)
= (𝑥 2 +1)2
𝑑𝑥
𝑑𝑦 𝑥 2 −2𝑥−1
= = (𝑥 2 − 2𝑥 − 1)(𝑥 2 + 1)−2
𝑑𝑥 (𝑥 2 +1)2
𝑑2𝑦
= (𝑥 2 − 2𝑥 − 1)(−2)(𝑥 2 + 1)−3 (2𝑥) + (𝑥 2 + 1)−2 (2𝑥 − 2)
𝑑𝑥 2
𝑑2𝑦
= (𝑥 2 + 1)−3 [−2𝑥(𝑥 2 − 2𝑥 − 1) + (𝑥 2 + 1)(𝑥 − 1)]
𝑑𝑥 2
𝑑2𝑦
= (𝑥 2 + 1)−3 [−2𝑥(𝑥 2 − 2𝑥 − 1) + (𝑥 2 + 1)(𝑥 − 1)]
𝑑𝑥 2
𝑑2𝑦
= (𝑥 2 + 1)−3 [−𝑥 3 + 3𝑥 2 + 3𝑥 − 1]
𝑑𝑥 2
𝑑2𝑦 𝑥 3 −3𝑥 2 −3𝑥+1
=−
𝑑𝑥 2 (𝑥 2 +1)3