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Let space space space space space straight y equals cube root of 2 straight x to the power of 4 plus straight x squared minus straight x end root equals left parenthesis 2 straight x to the power of 4 plus straight x squared minus straight x right parenthesis to the power of 1 third end exponent therefore space dy over dx equals 1 third left parenthesis 2 straight x to the power of 4 plus straight x squared minus straight x right parenthesis to the power of negative 2 over 3 end exponent. straight d over dx left parenthesis 2 straight x to the power of 4 plus straight x squared minus straight x right parenthesis space space space space space space space space space space space space space equals fraction numerator 1 over denominator 3 left parenthesis 2 straight x to the power of 4 plus straight x squared minus straight x right parenthesis to the power of bevelled 2 over 3 end exponent end fraction. left parenthesis 8 straight x cubed plus 2 straight x minus 1 right parenthesis equals fraction numerator left parenthesis 8 straight x cubed plus 2 straight x minus 1 right parenthesis over denominator 3 left parenthesis 2 straight x to the power of 4 plus straight x squared minus straight x right parenthesis to the power of bevelled 2 over 3 end exponent end fraction.

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162. Differentiate the following w.r.t.x:

fraction numerator 3 over denominator left parenthesis 2 straight x squared plus 5 right parenthesis squared end fraction

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163. Differentiate the following w.r.t.x:

square root of fraction numerator straight x squared minus 2 ax over denominator straight a squared minus 2 ab end fraction end root

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164. Differentiate the following w.r.t.x:

square root of fraction numerator 1 plus straight x over denominator 1 minus straight x end fraction end root

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165. Differentiate the following w.r.t.x:

square root of fraction numerator 1 minus straight x squared over denominator 1 plus straight x squared end fraction end root

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166.

Find f'(x) where f(x)= $square root of fraction numerator 1 minus straight x over denominator 2 plus straight x end fraction end root$

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167. Differentiate

square root of 3 straight x squared plus 2 straight x end root

w.r.t x.

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168. Differentiate

fraction numerator 1 over denominator straight x plus square root of 1 plus straight x squared end root end fraction

w.r.t x.

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169.

If space straight y equals open parentheses straight x plus square root of straight x squared plus straight a squared end root close parentheses to the power of straight n comma space prove space that space dy over dn equals fraction numerator ny over denominator square root of straight x squared plus straight a squared end root end fraction.

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170. Differentiate ^{x(1 + log x)}w.r.t.x.

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