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Continuity and Differentiability

Short Answer Type

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

straight e to the power of straight x cubed end exponent

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

square root of straight e to the power of square root of straight x end exponent end root comma straight X greater than 0

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153. Differentiate e^x + e^x² +....+ e^x⁵. w.r.t.x.

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154. Differentiate the following w.r.t.x :log (log x),x>1

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155. Differentiate the following w.r.t.x :log7 (log x)

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156. Differentiate (3x² - 9 x + 5)⁹ w.r.t.x.

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157. Differentiate

square root of 15 straight x squared minus straight x plus 1 end root straight w. straight r. straight t space straight x

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158. $Differentiate space square root of 3 straight x plus 2 end root plus fraction numerator 1 over denominator square root of 2 straight x squared plus 4 end root end fraction straight w. straight r. straight t space space straight x.$

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

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159. Differentiate

square root of fraction numerator straight x minus 1 over denominator straight x plus 2 end fraction end root

W.r.t x.

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

left parenthesis 4 straight x cubed minus 5 straight x squared plus 1 right parenthesis to the power of 4

91 Views