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

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Historical Background

Isaac Newton And Leibniz Discovered17th CenturyDerivative of Complex or Composite FunctionsLater used by Mathematicians, Engineers,

Chemists etc.

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Figure Source: www.channel4.com

Introduction To Chain Rule

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Electrical Engineering Field

Power Calculation In Electrical Circuits

Figure Source: ENGR 1990 Engineering Mathematics

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Thank You

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Queries

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