Product Rule +1

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Product Rule Dr.V.S.RaveendraNath M.Sc.,M.Ed.,P.hd.

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2/13/2012 VSR 2 Product Rule. The product Rule may also be stated as follows: Derivative of the product of two functions = 1st function × derivative of 2nd function + 2nd function × derivative of 1st function.

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2/13/2012 VSR 3 Q.1. Differentiate using product rule (3x + 1) ( 2x – 5) Answer We have (1) (2) = (3x + 1). 2 + (2x – 5 ) . 3 = 6x + 2 + 6x – 15 = 12x – 13

Q.2. Differentiate f(x) = (x – 1) (x – 2) : 

2/13/2012 VSR 4 Q.2. Differentiate f(x) = (x – 1) (x – 2) Answer We have

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2/13/2012 VSR 5 Q.3. Differentiate f(x) = (x2 – 4x + 5) ( x2 – 2) Answer We have

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2/13/2012 VSR 6 Q.4. Differentiate f(x) = (x3 – 3x2 + 4) (4x5 + x2 – 1 ) Answer Do it by yourself

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2/13/2012 VSR 7 Important Results. Trigonometric Derivatives Derivative of sin x = cos x ” cos x = - sin x ” tan x = sec2 x ” cot x = - cosec2 x ” sec x = sec x. tan x ” cosec x = - cosec x . cot x

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2/13/2012 VSR 8 Important Results. Trigonometric Derivatives Derivative of ex = ex ” log x = 1/x ” ”

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2/13/2012 VSR 9 Q.5. Differentiate ( 1 + x2 ) cos x Answer We have

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2/13/2012 VSR 10 Q.6. Differentiate (x2 + x ) cosec x Answer We have

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2/13/2012 VSR 11 Q.7. Differentiate ex cot x Answer We have

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2/13/2012 VSR 12 Q.7. Differentiate f(x) = x sin x.log x Answer We have

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2/13/2012 VSR 13 Q.8. Differentiate f(x) = x3 ex log x. Answer We have

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2/13/2012 VSR 14 Q.9. Differentiate f(x) = ( 1 + 2 tan x )(5 + 4 cos x) Answer By product rule we have

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2/13/2012 VSR 15 Arranged as this From this take 8/cosx out

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2/13/2012 VSR 16 The End of Product Rule.

Quotient Rule. : 

2/13/2012 VSR 17 Quotient Rule. Derivative of a quotient = Denominator × derivative of numerator - numerator × derivative of denominator.  (denominator)2

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2/13/2012 VSR 18 Q.1. Find f(x) = by quotient rule. Answer We have

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2/13/2012 VSR 19 Q.2. Find f(x) = by quotient rule. Answer Do it by yourself

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2/13/2012 VSR 20 Q.3. Differentiate f(x) = Answer We have

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2/13/2012 VSR 21 Q.4. Differentiate f(x) = We have

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2/13/2012 VSR 22 =1

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2/13/2012 VSR 23 Q.5. Differentiate f(x) = Answer We have

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2/13/2012 VSR 24

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2/13/2012 VSR 25 Q.6. Differentiate w.r.t. x Do it by yourself Answer

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2/13/2012 VSR 26 Q.7. Differentiate f(x) = We have Answer

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2/13/2012 VSR 27

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2/13/2012 VSR 28 Q.8. We know that differentiate 1/x by quotient rule and verify that the answer is the same. By quotient rule we know , Answer Hence the answer is the same.

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2/13/2012 VSR 29 The End of the Quotient Rule. The End of the Quotient Rule. The End of the Quotient Rule.

The Chain Rule Or Function of a Function Rule. : 

2/13/2012 VSR 30 The Chain Rule Or Function of a Function Rule.

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2/13/2012 VSR 31 U – Formulae

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2/13/2012 VSR 32

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2/13/2012 VSR 33

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2/13/2012 VSR 34

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2/13/2012 VSR 35 Q.1. By chain rule find Let y = f(x) = u6, where u = x + 1 By chain rule Answer

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2/13/2012 VSR 36 Q.2. By chain rule find f(x) = (2x+5)2 Let y = f(x) = u2, where u = 2x + 5 By chain rule Answer

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2/13/2012 VSR 37 Q.3. By chain rule find f(x) = (x - 1)-3 Answer Let y = f(x) = u-3, where u = x - 1 By chain rule

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2/13/2012 VSR 38 Q.4. By chain rule find 1). f(x) = (1 – 5x)4 2). f(x) = (1 – 2x)2 3). f(x) = (3x2 + 2)2 Do it by yourself Answer 1). - 20(1 – 5x)3 2). - 4(1 – 2x) 3). 12x(3x2 + 2)

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2/13/2012 VSR 39 Q5. Use chain rule find the derivatives of the function Answer Let y = f(x) = u2, where u =

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2/13/2012 VSR 40 By using quotient rule

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2/13/2012 VSR 41 Hence the chain rule

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2/13/2012 VSR 42

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2/13/2012 VSR 43 Q.6. Differentiate the function w.r.t.x f(x) = Answer Let y = f(x) = u , where u = 1 – 2x Hence the chain rule

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2/13/2012 VSR 44

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2/13/2012 VSR 45 Q.9. Find the derivatives of the function Answer We have

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2/13/2012 VSR 46 Q.9. Find dy/dx if x2 + y2 = r2 Answer We have x2 + y2 = r2 Differtiating w.r.t.x through, we get,

The End of the Quotient Rule. : 

The End of the Quotient Rule The End of the Quotient Rule The End of the Quotient Rule The End of the Quotient Rule The End of the Quotient Rule The End of the Quotient Rule The End of the Quotient Rule The End of the Quotient Rule 2/13/2012 VSR 47 The End of the Quotient Rule. The End of the Chapter Limit and Derivatives. The End of the Chapter Limits and Derivatives.