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

space fraction numerator sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction

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

cos 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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413. Differentiate the following functions w.r.t.x:

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

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

sin to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator square root of 1 plus straight x squared end root end fraction close parentheses

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415. Differentiate the following functions w.r.t.x

fraction numerator cos left parenthesis log space straight x right parenthesis plus tan to the power of negative 1 end exponent straight x over denominator straight e to the power of straight x end fraction

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

fraction numerator straight e to the power of straight x squared end exponent space tan to the power of negative 1 end exponent straight x over denominator square root of 1 plus straight x squared end root end fraction

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

fraction numerator cos to the power of negative 1 end exponent begin display style straight x over 2 end style over denominator square root of 2 straight x plus 7 end root end fraction comma space minus 2 less than straight x less than 2

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418. Differentiate :

fraction numerator straight x space cos to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction comma space straight x not equal to 1 comma space straight w. straight r. straight t. straight x

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Let space space space space space straight y equals cot to the power of negative 1 end exponent straight x plus cot to the power of negative 1 end exponent 1 over straight x therefore space dy over dx equals negative fraction numerator 1 over denominator 1 plus straight x squared end fraction minus fraction numerator 1 over denominator 1 plus begin display style 1 over straight x squared end style end fraction. straight d over dx open parentheses 1 over straight x close parentheses equals negative fraction numerator 1 over denominator 1 plus straight x squared end fraction minus fraction numerator straight x squared over denominator 1 plus straight x squared end fraction cross times open parentheses negative 1 over straight x squared close parentheses space space space space space space space space space space space space space equals negative fraction numerator 1 over denominator 1 plus straight x squared end fraction plus fraction numerator 1 over denominator 1 plus straight x squared end fraction therefore space straight d over dx left parenthesis straight y right parenthesis equals 0 space rightwards double arrow space straight y equals constant therefore space cot to the power of negative 1 end exponent straight x plus cot to the power of negative 1 end exponent 1 over straight x space space is space constant

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420.

If space straight y equals fraction numerator 2 over denominator square root of straight a squared minus straight b squared end root end fraction tan to the power of negative 1 end exponent open parentheses square root of fraction numerator straight a minus straight b over denominator straight a plus straight b end fraction end root tan straight x over 2 close parentheses comma space prove space that space dy over dx equals fraction numerator 1 over denominator straight a plus straight b space cos space straight x end fraction comma space straight a greater than straight b greater than 0.

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