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Limits and Derivatives

Short Answer Type

61. Find the derivatives of the following :

space space space space space space space space space space space space space space space space space space space fraction numerator ax squared plus bx plus straight c over denominator px squared plus qx plus straight r end fraction

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62.

Find $space space space space space space space space space space space space space space space space space space space space dy over dx space space if space left parenthesis straight i right parenthesis space straight y space space equals space fraction numerator 1 minus square root of straight x over denominator 1 plus square root of straight x end fraction$

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63.

Find $space space space space space space space space space space space space space space space space space space space space space space space space space space space space dy over dx space space if space left parenthesis straight i right parenthesis space straight y space space equals space fraction numerator straight x squared minus straight x plus 1 over denominator straight x squared plus straight x plus 1 end fraction$

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64.
Derivative of tan x :

Let f(x) =, tan x, Therefore, $space space space space space space space space space space space space space space space space straight f left parenthesis straight x right parenthesis space equals space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction$

$space space space space space space space space space space space space space space space equals space space limit as straight h rightwards arrow 0 of fraction numerator tan left parenthesis straight x plus straight h right parenthesis minus tanx over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator begin display style fraction numerator sin left parenthesis straight x plus straight h right parenthesis over denominator cos left parenthesis straight x plus straight h right parenthesis end fraction end style minus begin display style fraction numerator sin space straight x over denominator cos space straight x end fraction end style over denominator straight h end fraction$

$space space space space space space space space space space space space space space space space space space space space space space equals space space limit as straight h rightwards arrow 0 of fraction numerator sin left parenthesis straight x plus straight h right parenthesis cps space straight x minus space sin space straight x space cos space left parenthesis straight x plus straight h right parenthesis over denominator cos left parenthesis straight x plus straight h right parenthesis. cos space straight x. straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator sin left parenthesis straight x plus straight h minus straight x right parenthesis over denominator straight h space cos space straight x. space cos space left parenthesis straight x plus straight h right parenthesis end fraction$

$space space space space space space space space space space space space space equals space space space limit as straight h rightwards arrow 0 of fraction numerator sin space space straight h over denominator straight h end fraction. space limit as straight h rightwards arrow 0 of fraction numerator 1 over denominator cos space straight x. space cos space left parenthesis straight x plus straight h right parenthesis end fraction$

$space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space space space 1. fraction numerator 1 over denominator cos space straight x space. space cos space straight x end fraction equals space fraction numerator 1 over denominator cos squared straight x end fraction equals space space space sec squared straight x space space space space space space$ $space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space limit as straight x rightwards arrow 0 of fraction numerator sin space straight x over denominator straight x end fraction equals 1 close square brackets$

So, $space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight d over dx left parenthesis tan space straight x right parenthesis space equals space sec squared straight x space for space all space straight x space element of space straight R space open parentheses straight R space minus space odd space multiples space of space straight pi over 2 close parentheses$

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65. Derivative of cot x :

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66. Derivative of sec x :

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67. Derivative of cosec x :

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68. Find the derivatives of the following functions from first principle :

sin (x+1)

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69. Find the derivatives of the following functions from first principle :

$space space space space space space space space space space space space space space open parentheses straight x minus straight pi over 8 close parentheses$