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Continuity and Differentiability

Short Answer Type

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

cube root of 2 straight x to the power of 4 plus straight x squared minus straight x end root

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

Let space space space space space space straight y equals square root of fraction numerator 1 minus straight x squared over denominator 1 plus straight x squared end fraction end root equals open parentheses fraction numerator 1 minus straight x squared over denominator 1 plus straight x squared end fraction close parentheses to the power of 1 half end exponent therefore space dy over dx equals 1 half open parentheses fraction numerator 1 minus straight x squared over denominator 1 plus straight x squared end fraction close parentheses to the power of negative 1 half end exponent straight d over dx open parentheses fraction numerator 1 minus straight x squared over denominator 1 plus straight x squared end fraction close parentheses space space space space space space space space space space space space space equals 1 half open parentheses fraction numerator 1 plus straight x squared over denominator 1 minus straight x squared end fraction close parentheses to the power of 1 half end exponent. open square brackets fraction numerator left parenthesis 1 plus straight x squared right parenthesis. left parenthesis negative 2 straight x right parenthesis minus left parenthesis 1 minus straight x squared right parenthesis.2 straight x over denominator left parenthesis 1 plus straight x squared right parenthesis squared end fraction close square brackets space space space space space space space space space space space space space equals fraction numerator left parenthesis 1 plus straight x squared right parenthesis to the power of begin display style 1 half end style end exponent over denominator 2 left parenthesis 1 minus straight x squared right parenthesis to the power of 1 half end exponent end fraction. fraction numerator negative 2 straight x minus 2 straight x cubed minus 2 straight x plus 2 straight x cubed over denominator left parenthesis 1 plus straight x squared right parenthesis squared end fraction space space space space space space space space space space space space space equals fraction numerator left parenthesis 1 plus straight x squared right parenthesis to the power of begin display style 1 half end style end exponent over denominator 2 left parenthesis 1 minus straight x squared right parenthesis to the power of 1 half end exponent end fraction cross times fraction numerator negative 4 straight x over denominator left parenthesis 1 plus straight x squared right parenthesis squared end fraction equals negative fraction numerator 2 straight x over denominator left parenthesis 1 minus straight x squared right parenthesis to the power of bevelled 1 half end exponent left parenthesis 1 plus straight x squared right parenthesis to the power of bevelled 3 over 2 end exponent end fraction

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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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