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Continuity and Differentiability

Short Answer Type

401. Differentiate the following w.r.t.x: $sin left parenthesis tan to the power of negative 1 end exponent space straight e to the power of negative straight x end exponent right parenthesis$

Let space space space space space straight y equals sin left parenthesis tan to the power of negative 1 end exponent space straight e to the power of negative straight x end exponent right parenthesis therefore space dy over dx equals cos left parenthesis tan to the power of negative 1 end exponent space straight e to the power of negative straight x end exponent right parenthesis. straight d over dx left parenthesis tan to the power of negative 1 end exponent space straight e to the power of negative straight x end exponent right parenthesis space space space space space space space space space space space space space equals cos left parenthesis tan to the power of negative 1 end exponent space straight e to the power of negative straight x end exponent right parenthesis. fraction numerator 1 over denominator 1 plus left parenthesis straight e to the power of negative straight x end exponent right parenthesis squared end fraction. straight d over dx left parenthesis straight e to the power of negative straight x end exponent right parenthesis space space space space space space space space space space space space space equals cos left parenthesis tan to the power of negative 1 end exponent space straight e to the power of negative straight x end exponent right parenthesis. fraction numerator 1 over denominator 1 plus left parenthesis straight e to the power of negative straight x end exponent right parenthesis squared end fraction. straight e to the power of negative straight x end exponent. straight d over dx left parenthesis negative straight x right parenthesis space space space space space space space space space space space space space equals cos left parenthesis tan to the power of negative 1 end exponent space straight e to the power of negative straight x end exponent right parenthesis. fraction numerator 1 over denominator 1 plus left parenthesis straight e to the power of negative straight x end exponent right parenthesis squared end fraction. straight e to the power of negative straight x end exponent. left parenthesis negative 1 right parenthesis space space space space space space space space space space space space space equals negative fraction numerator straight e to the power of negative straight x end exponent over denominator 1 plus straight e to the power of negative 2 straight x end exponent end fraction. cos left parenthesis tan to the power of negative 1 end exponent straight e to the power of negative straight x end exponent right parenthesis.

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402. Differentiate the following function w.r.t.x:

cot to the power of negative 1 end exponent square root of straight x

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403. Differentiate the following function w.r.t.x:

sin minus 1 left parenthesis cos space straight x right parenthesis comma space 0 less than straight x less than straight pi over 2

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404. Differentiate the following function w.r.t.x:

cos minus 1 open parentheses fraction numerator straight x over denominator straight x plus 1 end fraction close parentheses

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405. Differentiate the following function w.r.t.x:

tan to the power of negative 1 end exponent open parentheses fraction numerator cos space straight x over denominator 1 plus sin space straight x end fraction close parentheses

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406. Differentiate the following function w.r.t.x:

left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis squared

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407. Differentiate the following function w.r.t.x:

straight e to the power of cot to the power of negative 1 end exponent straight x squared end exponent

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408. Differentiate the following function w.r.t.x:

straight x space sec to the power of negative 1 end exponent straight x

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409. Differentiate the following function w.r.t.x:

tan to the power of negative 1 end exponent left parenthesis log space straight x right parenthesis

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410. Differentiate the following functions w.r.t.x:

cot to the power of negative 1 end exponent square root of fraction numerator 1 plus straight x over denominator 2 end fraction end root

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