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Limits and Derivatives

Short Answer Type

71.

Find the derivatives of the following functions :

5 sec x + 4 cos x

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72.

Find the derivatives of the following functions :

sin x cos x

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73.

Find the derivatives of the following functions :

2 tan x - 7 sec x

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74. Find the derivatives of the following functions :

space space space space space space space space fraction numerator sec space straight x space minus 1 over denominator sec space straight x space plus 1 end fraction

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75. Find the derivatives of the following functions :

space space space space fraction numerator sin left parenthesis straight x plus straight a right parenthesis over denominator cos space straight x end fraction

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76. Find the derivatives of the following functions :

(ax²+sin x) (p + q cos x)

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77. Find the derivatives of the following functions :

"<pre

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78. Find the derivatives of the following functions :

(x + sec x)(x - tan x)

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79. Find the derivatives of the following functions :

$space space space space space space space space space space space space space space space space space space fraction numerator straight x over denominator 1 plus tan space straight x end fraction$

Let $space space space space space space space space space space space space fraction numerator straight x over denominator 1 plus tan space straight x end fraction$

Differentiating both sides w.r.t. x, we get

$space space space space space space space space space space space space space space space space space space space space space space space dy over dx equals fraction numerator left parenthesis 1 plus tan space straight x right parenthesis begin display style straight d over dx end style straight x minus straight x begin display style straight d over dx end style left parenthesis 1 plus tan space straight x right parenthesis over denominator blank end fraction equals space fraction numerator left parenthesis 1 plus tan space straight x right parenthesis space 1 minus straight x left parenthesis 0 plus sec squared straight x right parenthesis over denominator left parenthesis 1 plus tan space straight x right parenthesis squared end fraction$

$space space space space space space space space space space space space space space space space space equals space space fraction numerator 1 plus tan space straight x space minus space straight x space sec squared straight x over denominator left parenthesis 1 plus tan space straight x right parenthesis squared end fraction$

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80.

Evaluate the following limits :

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