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Limits and Derivatives

Short Answer Type

41. Find the derivative of the following :

space space space space space space space space fraction numerator 2 over denominator straight x plus 1 end fraction minus fraction numerator straight x squared over denominator 3 straight x minus 1 end fraction

105 Views

42. Find the derivative of the following :

space space space space space space space fraction numerator 1 plus begin display style 1 over straight x end style over denominator 1 minus begin display style 1 over straight x end style end fraction

110 Views

43. Find the derivatives of the following :

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102 Views

44. Find the derivatives of the following :

(ax+b)ⁿ(cx+ d)ⁿ

Let y = (ax+b)ⁿ(cx+ d)ⁿ

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

$space space space space space space space space space dy over dx equals straight d over dx left square bracket left parenthesis ax plus straight b right parenthesis to the power of straight n space left parenthesis cx plus space straight d right parenthesis to the power of straight n right square bracket$

 $space space space space space space space space space space space space space space space space space equals space space space space left parenthesis ax space plus straight b right parenthesis to the power of straight n straight n left parenthesis cx plus straight d right parenthesis to the power of straight n minus 1 space end exponent straight c space plus space left parenthesis cx plus straight d right parenthesis to the power of straight n space straight d over dx left parenthesis ax plus straight b right parenthesis to the power of straight n$

 $space space space space space space space equals space space space space left parenthesis ax plus straight b right parenthesis to the power of straight n straight n left parenthesis cx plus straight d right parenthesis to the power of straight n minus 1 end exponent straight c plus left parenthesis cx plus straight d right parenthesis to the power of straight n space straight n space left parenthesis ax plus straight b right parenthesis to the power of straight n minus 1 end exponent a$

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 $space space space space space space space space space space space space equals space straight n open parentheses ax plus straight b close parentheses to the power of straight n minus 1 end exponent open parentheses cx plus straight d close parentheses to the power of straight n minus 1 end exponent open parentheses 2 ax plus bc plus ad close parentheses$