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

Let space space space space space space space square root of straight e to the power of square root of straight x end exponent end root equals open parentheses straight e to the power of square root of straight x end exponent close parentheses to the power of 1 half end exponent therefore space space space space space dy over dx equals 1 half open parentheses straight e to the power of square root of straight x end exponent close parentheses to the power of 1 half end exponent. straight d over dx open parentheses straight e to the power of square root of straight x end exponent close parentheses equals fraction numerator 1 over denominator 2 square root of straight e to the power of square root of straight x end exponent end root end fraction. straight e to the power of square root of straight x end exponent. straight d over dx open parentheses square root of straight x close parentheses space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 square root of straight e to the power of square root of straight x end exponent end root end fraction. straight e to the power of square root of straight x end exponent. fraction numerator 1 over denominator 2 square root of straight x end fraction equals 1 fourth fraction numerator straight e to the power of square root of straight x end exponent over denominator square root of straight x space straight e to the power of square root of straight x end exponent end root end fraction

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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

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