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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

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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

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43. Find the derivatives of the following :

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44. Find the derivatives of the following :

(ax+b)ⁿ(cx+ d)ⁿ

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45. Derivative of xⁿ.

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46.
Derivative of (ax + b)ⁿ, n being a rational number.

Let f(x) = (ax+b)ⁿ

$space space space space space space space space space space space space straight f left parenthesis straight x right parenthesis space equals space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator open square brackets straight a open parentheses straight x plus straight h close parentheses plus straight b close square brackets to the power of straight n minus left parenthesis ax plus straight b right parenthesis to the power of straight n over denominator straight h end fraction$

$space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of fraction numerator left square bracket ax plus straight b plus ha right square bracket to the power of straight n minus left parenthesis ax plus straight b right parenthesis to the power of straight n over denominator straight h end fraction$

$space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of left parenthesis ax plus straight b right parenthesis to the power of straight n fraction numerator open curly brackets open parentheses 1 plus begin display style fraction numerator ah over denominator ax plus straight b end fraction end style close parentheses to the power of straight n minus 1 close curly brackets over denominator straight h end fraction$

$space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis a x plus b right parenthesis to the power of n. limit as h rightwards arrow 0 of 1 over h open curly brackets open parentheses 1 plus fraction numerator n h a over denominator a x plus b end fraction plus fraction numerator n left parenthesis n minus 1 right parenthesis over denominator left enclose bottom enclose 2 end enclose end fraction. fraction numerator h squared a squared over denominator left parenthesis a x plus b right parenthesis squared end fraction plus..... close parentheses minus 1 close curly brackets$

$space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis ax plus straight b right parenthesis to the power of straight n. limit as straight h rightwards arrow 0 of open curly brackets fraction numerator nha over denominator ax plus straight b end fraction plus fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator bottom enclose left enclose 2 end enclose end fraction. fraction numerator straight h squared straight a squared over denominator left parenthesis ax plus straight b right parenthesis squared end fraction plus...... close curly brackets$

$space space space space space space space space space space space space equals space left parenthesis ax plus straight b right parenthesis to the power of straight n. limit as straight h rightwards arrow 0 of 1 over straight h open curly brackets fraction numerator na over denominator ax plus straight b end fraction plus fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator bottom enclose left enclose 2 end enclose end fraction. fraction numerator ha squared over denominator left parenthesis ax plus straight b right parenthesis squared end fraction plus...... close curly brackets$

$space space space space space space space space equals space left parenthesis ax plus straight b right parenthesis to the power of straight n open parentheses fraction numerator na over denominator ax plus straight b end fraction plus 0 close parentheses equals na left parenthesis ax plus straight b right parenthesis to the power of straight n minus 1 end exponent$

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

Corollary : Put a = 1m, b = 0

$space space space space space space straight d over dx left parenthesis straight x to the power of straight n right parenthesis space equals space nx to the power of straight n minus 1 end exponent$