Differentiation Techniques

Ravinder Kumar, Jun 5, 2019

This book is about techniques of differentiation, namely differentiation by 1. Constant Times Rule 2. Sum/Difference Rule 3. Product Rule 4. Quotient Rule 5. Chain Rule 6. Implicit Differentiation Rule An applet on implicit differentiation by Tim Brezinski is included with thanks and due credit to him to make the book complete. For more on derivative, techniques of differentiation, and applications of derivative check my free online calculus book "Flipped Classroom Calculus of Single Variable" at https://versal.com/learn/vh45au/

Table of Contents

  1. Linearity Rule of Derivatives
    1. Constant Times, Sum/Diff Rules
  2. Product and Quotient Rules
    1. Product Rule of Differentiation
    2. Quotient Rule of Differentiation
    3. Product and Quotient Rules of Differentiation
  3. Chain Rule
    1. Chain Rule With One Inner Function
    2. Chain Rule with Two Inner Functions
  4. Implicit Differentiation
    1. Explore Implicitely Defined Graphs
    2. Implicit Differentiation: Intro
    3. Implicit Differentiation

1.

Linearity Rule of Derivatives

  • 1. Constant Times, Sum/Diff Rules

1.1.

Constant Times, Sum/Diff Rules

Derivative of k times a function is k times derivative of the function. Thus[br] [math]\frac{d}{dx}\left(kf\left(x\right)\right)=k\frac{d}{dx}\left(f\left(x\right)\right)[/math][br]Sum/Difference of the derivative of two functions is sum/difference of the derivatives of the two functions. Thus [br] [math]\frac{d}{dx}\left(f\left(x\right)\pm g\left(x\right)\right)=\frac{d}{dx}\left(f\left(x\right)\right)\pm\frac{d}{dx}\left(g\left(x\right)\right)[/math][br][br]These two rules show that the process of finding derivatives is linear. Namely,[br][br] [math]\frac{d}{dx}\left(kf\left(x\right)+lg\left(x\right)\right)=k\frac{d}{dx}\left(f\left(x\right)\right)+l\frac{d}{dx}\left(g\left(x\right)\right)[/math][br][br]The purpose of this applet is to facilitate practice of this rule.
– Ravinder Kumar

2.

Product and Quotient Rules

The product rule of differentiation may be applied when the function to be differentiated is a product of two functions: [math](f(x) g(x))^{'}=f^{'}(x) g(x)+f(x) g^{'}(x)[/math] The quotient rule of differentiation may be applied when the function to be differentiated is a quotient of two functions: [math]\left(\frac{f(x)}{g(x)}\right)'=\frac{f'(x)g(x)-f(x)g'(x)}{(g(x))^2}[/math]

  • 1. Product Rule of Differentiation
  • 2. Quotient Rule of Differentiation
  • 3. Product and Quotient Rules of Differentiation

2.1.

Product Rule of Differentiation

Purpose of this applet is to provide practice on product rule of differentiation, which is [br][math]\left(uv\right)'=u'v+uv'[/math][br]An example is given for help. There are plenty of questions for practice. Follow the instructions which can be viewed by checking the checkbox Instructions.
– Ravinder Kumar

2.2.

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2.3.

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3.

Chain Rule

Chain Rule of derivative is applied for the derivative of a function of a function, such as [math]sin(x^{11})[/math] or [math]e^{cos(x)}[/math]. If f is a function of t and t is a function of x, then [math]\frac{d}{dx}(f(t))=\frac{d}{dt}(f(t))\frac{d}{dx}(t(x))[/math] In this situation, sometimes, f(t) is called outer function and t(x) is called inner function.

  • 1. Chain Rule With One Inner Function
  • 2. Chain Rule with Two Inner Functions

3.1.

Chain Rule With One Inner Function

This applet can be used to find derivative of a composite function generated with one inner function (composition of two functions). Enter Inner function in [i]x[/i] and outer function in [i]t[/i]. You are advised to solve the problem using pen and paper before comparing with the solution given by this applet. Please note that the simplification produced in the applet may differ from yours and may not look neat sometimes. Use this applet for practice on chain rule with one inner function.
You should have studied differentiating constant times, sum/difference, product, and quotient rules of differentiation and feel fairly comfortable with them.[br][br]For more on chain rule check my online book [br]"Flipped Classroom Calculus of Single Variable"[br]https://versal.com/learn/vh45au/
– Ravinder Kumar

3.2.

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4.

Implicit Differentiation

When a function or expression is expressed in terms of, say x and y, different from [math]y=f(x)[math], it may be called an implicit form of the expression, for example, [math]3x+y=7[/math], [math]y^2+4=2x \sin(x+y) -3[/math] In this case the method to find [math]\frac{dy}{dx}[/math] or [math]\frac{dx}{dy}[/math] is called implicit differentiation.

  • 1. Explore Implicitely Defined Graphs
  • 2. Implicit Differentiation: Intro
  • 3. Implicit Differentiation

4.1.

Explore Implicitely Defined Graphs

Graphs are said to be implicitely defined when their equation is not in the form [math]y=f\left(x\right)[/math] but rather in the form [math]f\left(x,y\right)=c[/math], where c may be zero.[br][br]For more on implicit differentiation check my online book [br]"Flipped Classroom Calculus of Single Variable"[br]https://versal.com/learn/vh45au/
– Ravinder Kumar

4.2.

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4.3.

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Information