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Continuity And Differentiability

Question

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$Differentiate space log left parenthesis 1 plus straight theta right parenthesis space straight w. straight r. straight t. space sin to the power of negative 1 end exponent space straight theta$

Solution

Let space space space space space straight y equals log left parenthesis 1 plus straight theta right parenthesis comma space straight u equals sin to the power of negative 1 end exponent space straight theta therefore space dy over dθ equals fraction numerator 1 over denominator 1 plus straight theta end fraction comma space du over dθ equals fraction numerator 1 over denominator square root of 1 minus straight theta squared end root end fraction therefore space dy over du equals fraction numerator 1 over denominator 1 plus straight theta end fraction cross times fraction numerator square root of 1 minus straight theta squared end root over denominator 1 plus straight theta end fraction equals fraction numerator square root of 1 minus straight theta squared end root over denominator 1 plus straight theta end fraction

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