Higher order derivatives and partial derivatives are discussed. Higher order derivatives refer to taking successive derivatives of a function, such as the second derivative being the derivative of the first derivative. Partial derivatives refer to taking the derivative of a function of multiple variables with respect to one of the variables, holding the other variables constant. Examples are provided of finding higher order derivatives and partial derivatives of various functions.

1. Higher Derivatives &
Partial Differentiation
Raymund T. de la Cruz
MAEd - Mathematics

2. Higher Derivatives
If y = f (x) is differentiable, its derivative y′ is also called the first derivative of f. If y′
is differentiable, its derivative is called the second derivative of f. If this second
derivative is differentiable, then its derivative is called the third derivative of f, and
so on.
First Derivative 𝑦′, 𝑓′ 𝑥 ,
𝑑𝑦
𝑑𝑥
, 𝐷 𝑥 𝑦
Second Derivative 𝑦′′, 𝑓′′ 𝑥 ,
𝑑2 𝑥
𝑑𝑦
, 𝐷 𝑥
2 𝑦
Third Derivative 𝑦′′′, 𝑓′′′ 𝑥 ,
𝑑3 𝑦
𝑑𝑥
, 𝐷 𝑥
3 𝑦
nth Derivative 𝑦(𝑛), 𝑓(𝑛) 𝑥 ,
𝑑 𝑛 𝑦
𝑑𝑥
, 𝐷 𝑥
𝑛 𝑦

3. Higher Derivatives
1. Find the second derivative of 𝑦 = 𝑥5 + 5𝑥4 − 10𝑥2 + 6
Solution:
𝑦′ = 5𝑥4 + 20𝑥3 − 20𝑥
𝑦′′ = 20𝑥3 + 60𝑥2 − 20
2. Find the third derivative of 𝑦 = (1 − 5𝑥)6
Solution:
𝑦′ = −5(6)(1 − 5𝑥)5
= −30 1 − 5𝑥 5
𝑦′′ = −5(5)(−30)(1 − 5𝑥)4
= 750(1 − 5𝑥)4
𝑦′′′ = −5(4)(750)(1 − 5𝑥)3
= −15000(1 − 5𝑥)3

4. Higher Derivatives
Find the indicated derivative.
1. 𝑦 = 3𝑥4 − 2𝑥2 + 𝑥 − 5 𝑦′′′
2. 𝑦 =
1
𝑥
𝑦 4
3. 𝑓 𝑥 = 2 − 3𝑥2 𝑓′′ 𝑥
4. 𝑦 =
𝑥
𝑥−1
{𝑦′′}
1. 𝑦′′ = 72𝑥
2. 𝑦 4 =
105
16𝑥9 2
3. 𝑓′′ 𝑥 = −
6
(2−3𝑥2)3 2
4. 𝑦′′ =
4−𝑥
4(𝑥−1)5 2

5. Partial Differentiation
Let 𝑧 = 𝑓 𝑥, 𝑦 be a function of two variables. If 𝑥 varies while 𝑦 is held fixed, 𝑧
becomes a function of 𝑥. Then its derivative with respect to 𝑥 is called partial
derivative of 𝑓 with respect to 𝑥 and is denoted 𝑓𝑥 𝑥, 𝑦 𝑜𝑟
𝑑𝑧
𝑑𝑥
𝑜𝑟
𝑑𝑓
𝑑𝑥
. Similarly, if 𝑦
varies while 𝑥 is held fixed, the partial derivative of 𝑓 with respect to 𝑦 and is
denoted 𝑓𝑦 𝑥, 𝑦 𝑜𝑟
𝑑𝑧
𝑑𝑦
𝑜𝑟
𝑑𝑓
𝑑𝑦
.
1. Find the partial derivatives of 𝑓 𝑥, 𝑦 = 𝑥2 sin 𝑦.
Solution:
𝑓𝑥 𝑥, 𝑦 = 2𝑥 sin 𝑦
𝑓𝑦 𝑥, 𝑦 = 𝑥2 cos 𝑦

6. Partial Differentiation
2. Find the partial derivatives of 𝑓 𝑥, 𝑦 = 𝑥2 + 3𝑥𝑦 + 𝑦2.
Solution:
𝑓𝑥 𝑥, 𝑦 = 2𝑥 + 3𝑦
𝑓𝑦 𝑥, 𝑦 = 3𝑥 + 2𝑦
3. Find the partial derivatives of 𝑓 𝑥, 𝑦, 𝑧 = 𝑥 cos(𝑦𝑧)
Solution:
𝑓𝑥 𝑥, 𝑦, 𝑧 = cos(𝑦𝑧)
𝑓𝑦 𝑥, 𝑦, 𝑧 = −𝑥𝑧 sin(𝑦𝑧)
𝑓𝑧 𝑥, 𝑦, 𝑧 = −𝑥𝑦 sin(𝑦𝑧)

7. Partial Differentiation
1. Find the partial derivatives of 𝑓 𝑥, 𝑦 =
𝑥
𝑦2 −
𝑦
𝑥2 .
2. Find the partial derivatives of 𝑓 𝑥, 𝑦 = sin 3𝑥 cos 4𝑦.
3. Find the partial derivatives of 𝑓 𝑥, 𝑦 = 2𝑥2 − 5𝑥𝑦 + 𝑦2.
4. For the given function of 𝑧, find
𝑑𝑧
𝑑𝑥
𝑎𝑛𝑑
𝑑𝑧
𝑑𝑦
, 𝑦𝑧 + 𝑥𝑧 + 𝑥𝑦 = 0.
1. 𝑓𝑥 𝑥, 𝑦 =
1
𝑦2 +
2𝑦
𝑥3 ; 𝑓𝑦 𝑥, 𝑦 = −
2𝑥
𝑦3 −
1
𝑥2
2. 𝑓𝑥 𝑥, 𝑦 = 3 cos 3𝑥 cos 4𝑦; 𝑓𝑦 𝑥, 𝑦 = −4 sin 3𝑥 sin 4𝑦
3. 𝑓𝑥 𝑥, 𝑦 = 4𝑥 − 5𝑦; 𝑓𝑦 = −5𝑥 + 2𝑦
4.
𝑑𝑧
𝑑𝑥
= −
𝑦+𝑧
𝑥+𝑦
;
𝑑𝑧
𝑑𝑦
= −
𝑥+𝑧
𝑥+𝑦