3. Second Order Partial Derivatives
The second-order (direct) partial derivative
signifies that the function has been differentiated
partially with respect to one of the independent
variables twice, while the other independent
variable has been held constant.
What Is Second-Order Partial Derivatives ?
5. EXAMPLE 1
Find the a) first b)second and c) cross partial derivatives for
𝑧 = 7𝑥3 + 9xy + 2𝑦5
Solution:
a)
𝜕𝑧
𝜕𝑥
= Zx = 21𝑥2+9y
𝜕𝑧
𝜕𝑦
= Zy = 9x+10𝑦4
b)
𝜕2 𝑧
𝜕𝑥2 =Zxx = 42x
𝜕2 𝑧
𝜕𝑦2 =Zyy = 40𝑦3
C) 𝜕2 𝑧
𝜕𝑦𝜕𝑥
=
𝜕
𝜕𝑦
𝜕𝑧
𝜕𝑥
=
𝜕
𝜕𝑦
(21𝑥2
+9y)= Zxy=9
𝜕2 𝑧
𝜕𝑥𝜕𝑦
=
𝜕
𝜕𝑥
𝜕𝑧
𝜕𝑦
=
𝜕
𝜕𝑥
(9x+ 10𝑦4
=)Zyx=9
6. EXAMPLE 1
Find the a) first b)second and c) cross partial
derivatives for 𝑧 = 3𝑥2 𝑦3 and evaluate below at x=4,y=1
Solution:
a) Zx = 6𝑥𝑦3 Zy=9𝑥2 𝑦2
Zx(4,1)=6(4)(1)3= 24 Zy(4,1)=9(4)2(1)2=144
b) Zxx=6𝑦3
Zyy=18𝑥2
𝑦
Zxx(4,1)=6(1)3
= 6 Zyy(4,1)=18 4 2
(1)=288
c) Zxy=
𝜕
𝜕𝑦
(6x𝑦3)=18x𝑦2 Zyx=
𝜕
𝜕𝑥
(9𝑥2 𝑦2)=18x𝑦2
Zxy(4,1)=18(4)(1)2= 72 Zyx 4,1 = 18(4)(1)2=72
7. Second Order Partial Derivatives
The second-order (direct) partial derivative signifies
that the function has been differentiated partially with
respect to one of the independent variables twice, while
the other independent variable has been held constant.
What Is Cross Partial Derivatives ?
8. Second Order Partial Derivatives
The second-order (direct) partial derivative
signifies that the function has been differentiated
partially with respect to one of the independent
variables twice, while the other independent
variable has been held constant.
What Is Second-Order Partial Derivatives ?
9. Second Order Partial Derivatives
The second-order (direct) partial derivative signifies
that the function has been differentiated partially with
respect to one of the independent variables twice, while
the other independent variable has been held constant.
What Is Cross Partial Derivatives ?
10. “
Quotations are commonly printed as a
means of inspiration and to invoke
philosophical thoughts from the reader.
