(a) Explanation Given information : The function g is differentiable. Formula used: Quotient Rule of differentiation, d d x [ f ( x ) g ( x ) ] = g ( x ) f ' ( x ) − f ( x ) g ' ( x ) [ g ( x ) ] 2 Proof : It is provided that the function g is differentiable. Quotient Rule of differentiation is, d d x [ f ( x ) g ( x ) ] = g ( x ) f ' ( x ) − f ( x ) g ' ( x ) [ g ( x ) ] 2 The Reciprocal Rule of differentiation is d d x [ 1 g ( x ) ] = − g ' ( x ) [ g ( x ) ] 2 . Take left hand side of above equation, d d x [ 1 g ( x ) ] when compared to left hand side of Quotient Rule, f ( x ) is 1 and g ( x ) remains same (b) To determine To calculate : The differentiation of the function y = 1 x 4 + x 2 + 1 with help of Reciprocal rule. (c) To determine To verify : The Power Rule of differentiation d d x ( x − n ) = − n x − n − 1 for all positive integers n with help of Reciprocal Rule.

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ISBN: 9781133109631

Author: Stewart, JAMES, Day, Troy

Publisher: Cengage Learning,

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Chapter 3.4, Problem 63E

(a)

To determine

To prove: The Reciprocal Rule of differentiation ddx[1g(x)]=−g'(x)[g(x)]2 if g is differentiable with help of Quotient Rule.

(b)

To determine

To calculate: The differentiation of the function y=1x4+x2+1 with help of Reciprocal rule.

(c)

To determine

To verify: The Power Rule of differentiation ddx(x−n)=−nx−n−1 for all positive integers n with help of Reciprocal Rule.

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Chapter 3.4, Problem 62E

Chapter 3.5, Problem 1E

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Chapter 3 Solutions

Biocalculus

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To prove : The Reciprocal Rule of differentiation d d x [ 1 g ( x ) ] = − g ' ( x ) [ g ( x ) ] 2 if g is differentiable with help of Quotient Rule .

Solution Summary: The author proves that the function g is differentiable with the help of Quotient Rule.

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To prove: The Reciprocal Rule of differentiation ddx[1g(x)]=−g'(x)[g(x)]2 if g is differentiable with help of Quotient Rule.

To calculate: The differentiation of the function y=1x4+x2+1 with help of Reciprocal rule.

To verify: The Power Rule of differentiation ddx(x−n)=−nx−n−1 for all positive integers n with help of Reciprocal Rule.

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