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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.

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

Let space space space space space straight y equals left parenthesis 4 straight x cubed minus 5 straight x squared plus 1 right parenthesis to the power of 4. therefore dy over dx equals 4 left parenthesis 4 straight x cubed minus 5 straight x squared plus 1 right parenthesis cubed. straight d over dx left parenthesis 4 straight x cubed minus 5 straight x squared plus 1 right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because straight d over dx left parenthesis straight u to the power of straight n right parenthesis equals straight n space straight u to the power of straight n minus 1 end exponent du over dx close square brackets space space space space space space space space space space space space equals 4 left parenthesis 4 straight x cubed minus 5 straight x squared plus 1 right parenthesis cubed. left parenthesis 12 straight x squared minus 10 straight x right parenthesis equals 8 straight x left parenthesis 4 straight x cubed minus 5 straight x squared plus 1 right parenthesis cubed. left parenthesis 6 straight x minus 5 right parenthesis

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