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Continuity and Differentiability

Short Answer Type

211. Differentiate x² e^3x w.r.t. (log x)².

Let space space straight y equals space straight x squared straight e to the power of 3 straight x end exponent space space and space straight u equals left parenthesis log space straight x right parenthesis squared Then space we space required space to space find space dy over du Now space dy over dx equals straight x squared. straight e to the power of 3 straight x end exponent.3 plus straight e to the power of 3 straight x end exponent.2 straight x equals straight e to the power of 3 straight x end exponent left square bracket 3 straight x squared plus 2 straight x right square bracket and space du over dx equals 2 left parenthesis log space straight x right parenthesis to the power of 1. straight d over dx left parenthesis log space straight x right parenthesis equals 2 space log.1 over straight x equals fraction numerator 2 space log space straight x over denominator straight x end fraction therefore space fraction numerator dy over denominator space space space du end fraction equals fraction numerator begin display style dy over dx end style over denominator begin display style du over dx end style end fraction equals straight e to the power of 3 straight x end exponent fraction numerator left parenthesis 3 straight x squared plus 2 straight x right parenthesis over denominator begin display style 2 over straight x log space straight x end style end fraction equals fraction numerator straight e to the power of 3 straight x end exponent left parenthesis 3 straight x squared plus 2 straight x squared right parenthesis over denominator 2 space log space straight x end fraction

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212.

Differentiate space log space straight x space straight w. straight r. straight t space 1 over straight x

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213.

Differentiate space straight x squared space straight w. straight r. straight t space straight x cubed

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214. Differentiate log (x e^x) w.r.t. x log x.

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215.

Find space dy over dx space if space straight x to the power of 6 plus straight y to the power of 6 plus 6 straight x squared straight y squared equals 16.

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216. Find

dy over dx

in the following :
x² + x y + y² = 100

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217. Find

dy over dx

in the following :
x³ + x² y + x y² + y³ = 81

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218.

Find space dy over dx space where space straight x squared over straight a squared plus straight y squared over straight b squared equals 2003

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219.

Find space dy over dx space where space 4 straight x squared plus 5 xy plus 7 straight y squared minus 8 straight x plus 9 straight y plus 3 equals 0.

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220.

Find space dy over straight d space where space straight x squared plus straight y squared equals 30 straight x space straight y

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Previous Year Papers

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Test Yourself

Short Answer Type

211. Differentiate x2 e3x w.r.t. (log x)2.

211. Differentiate x² e^3x w.r.t. (log x)².