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Continuity And Differentiability

Question

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If y=sin^-1(cos x)+cos^-1(sin x), find $dy over dx$ and express the result free from trigonometrical symbols.

Solution

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

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