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The Chain Rule(Three Variables Dependent)
Group - 4
Presented ToDr. Misbah Irshad
(Computer Science Department)1
Agenda of Today’s Presentation
Historical BackgroundComplete Introduction to Chain Rule
Different CasesExamplesTree Diagram
Applications of Chain Rule
2
Historical Background
Isaac Newton And Leibniz Discovered17th CenturyDerivative of Complex or Composite FunctionsLater used by Mathematicians, Engineers,
Chemists etc.
3
Figure Source: www.channel4.com
Introduction To Chain Rule
4
The Chain Rule
“A rule for differentiating a composite function (i.e. a function
depending upon another function) ”
Example:
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.
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
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
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.
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 ”
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
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.
Introduction To Chain Rule
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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
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
Introduction To Chain Rule
14
Three Variable Dependent Chain Rule
Case I
Example:
Examples
Formula Being Used
Introduction To Chain Rule
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Formulas Being Used
Case IIExamples
Introduction To Chain Rule
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Formulas Being Used
Implicit FunctionExamples
Applications Of Chain Rule
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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/
Applications Of Chain Rule
18
Electrical Engineering Field
Power Calculation In Electrical Circuits
Figure Source: ENGR 1990 Engineering Mathematics
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Thank You
20
Queries
Figure Source: www.media.giphy.com/media/SufoKsersIO2Y/giphy.gif
