These slides have a comprehensive explanation of chain rule for three independent variables x,y and z (i.e. w = f(x,y,z) ). The chain rule has been described with the help of examples.

1. The Chain Rule
(Three Variables Dependent)
Group - 4
Presented To
Dr. Misbah Irshad
(Computer Science Department) 1

2. Agenda of Today’s Presentation
Historical Background
Complete Introduction to Chain Rule
Different Cases
Examples
Tree Diagram
Applications of Chain Rule
2

3. Historical Background
Isaac Newton And Leibniz Discovered
17th Century
Derivative of Complex or Composite Functions
Later used by Mathematicians, Engineers,
Chemists etc.
3
Figure Source: www.channel4.com

4. Introduction To Chain Rule
4
The Chain Rule
“A rule for differentiating a composite function (i.e. a function
depending upon another function) ”
 Example:

5. The Chain Rule With Different Cases
5
 Starting From Earlier Knowledge
 Single Variable Dependent
“If z = f(x) is differentiable of function x, where x = g(t) is differentiable function
of t then z is differentiable function of t and ”
• Analogy; x is behaving like a chain between z and t.

6. The Chain Rule With Different Cases
6
 Starting From Earlier Knowledge
 Two Variable Dependent (a)
“If z = f(x, y) is differentiable of function x and y, where x = g(t) and y = h(t) are
differentiable function of s and t then z is differentiable function of t and”
• Analogy; x and y are behaving like chain between z and t.
Slide Source: James Stewart Calculus 8th Edition

7. The Chain Rule With Different Cases
7
 Starting From Earlier Knowledge
 Two Variable Dependent (b)
“If z = f(x, y) is differentiable of function x and y, where x = g(s, t) and y = h(s, t) are
differentiable function of s and t then z is differentiable function of s, t and ”
• Analogy; x and y are behaving like chain between z and s, t.
Figure Source: James Stewart Calculus 8th Edition

8. The Chain Rule With Different Cases
8
 Finally, Today’s Discussion
 Three Variable Dependent (a)
“If w = f(x, y, z) is differentiable of function x, y and z, where x = g(t), y = h(t) and z = b(t)
are differentiable function of t then w is differentiable function of t and ”
Slide Source: James Stewart Calculus 8th Edition
• Analogy; x , y and z are behaving like chain (connection) between w and t.

9. The Chain Rule With Different Cases
9
 Proof
 Three Variable Dependent
“If w = f(x, y, z) is differentiable of function x, y and z, where x = g(t), y = h(t) and z = b(t)
are differentiable function of t then w is differentiable function of t and ”

10. The Chain Rule With Different Cases
10
Three Variable Dependent (b)
“If w = f(x, y, z) is differentiable of function x, y and z, where x = g(s, t), y = h(s, t) and z = b(s, t)
are differentiable function of s and t then w is differentiable function of s and t and ”
• Analogy; x , y and z are behaving like chain (connection) between w and s, t.
Slide Source: James Stewart Calculus 8th Edition

11. The Chain Rule With Different Cases
11
Implicit Function
Three Variable Dependent
“Now we suppose that w is given implicitly as a function w = f(x, y, z) by an equation of the form
F(x,y,z,w) = 0 This means that F(x,y,z,f(x,w,z)) = 0 for all (x,y,z) in the domain of f . If F and f are
differentiable, then we can use the Chain Rule to differentiate the equation F(x,y,z,w) = 0 as follow;”
 Example of Implicit Function
A function or relation in which the dependent variable is not isolated on one of the equation.

12. Introduction To Chain Rule
12
 Memorizing Formulas
 Three Variable Dependent (a)
“If w = f(x, y, z) is differentiable of function x, y and z, where x = g(t), y = h(t) and z = b= (t) are
differentiable function of s and t then w is differentiable function of t and ”
Slide Source: James Stewart Calculus 8th Edition
Tree Diagram

13. Introduction To Chain Rule
13
 Memorizing Formulas
 Three Variable Dependent (b)
“If w = f(x, y, z) is differentiable of function x, y and z, where x = g(s, t), y = h(s, t) and z = b(s, t)
are differentiable function of s and t then w is differentiable function of s and t and ”
Slide Source: James Stewart Calculus 8th Edition
Tree Diagram

14. Introduction To Chain Rule
14
 Three Variable Dependent Chain Rule
 Case I
 Example:
Examples
Formula Being Used

15. Introduction To Chain Rule
15
Formulas Being Used
 Case II
Examples

16. Introduction To Chain Rule
16
Formulas Being Used
 Implicit Function
Examples

17. Applications Of Chain Rule
17
 Computer Science Field
 Artificial Intelligence + Speech Processing Courses (7th – 8th
Semester ITU)
 Speech Reorganization; Neural Networks & Back Propagation
Algorithms (Course Topic)
Figure Source and For Further Study: www.jeremykun.com/2012/12/09/neural-networks-and-backpropagation/

18. Applications Of Chain Rule
18
 Electrical Engineering Field
 Power Calculation In Electrical Circuits
Figure Source: ENGR 1990 Engineering Mathematics

19. 19
Thank You

20. 20
Queries
Figure Source: www.media.giphy.com/media/SufoKsersIO2Y/giphy.gif