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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.

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47. Derivative of sin x :

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48. Find the derivatives of the following functions :

x¹⁰⁰

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49. Find the derivatives of the following functions :

6x¹⁰⁰- x⁵⁵+ x

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50. Find the derivatives of the following functions :

$space space space fraction numerator 1 over denominator square root of straight x end fraction$

Let $space space space space space space space space straight y space equals space fraction numerator 1 over denominator square root of straight x end fraction equals straight x to the power of negative 1 half end exponent$

∴ $space space space space space space space space space dy over dx equals straight d over dx left parenthesis straight x to the power of negative 1 half end exponent right parenthesis space equals space minus 1 half straight x to the power of negative 1 half minus 1 end exponent$ $space space space space space space open square brackets because straight d over dx left parenthesis straight x to the power of straight n right parenthesis equals space nx to the power of straight n minus 1 end exponent close square brackets$

space space space space space space space equals space space minus 1 half straight x to the power of negative 3 over 2 end exponent

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