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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $log open parentheses straight x plus 2 plus square root of straight x squared plus 4 straight x plus 1 end root close parentheses$

Solution

Let space straight y equals log open square brackets straight x plus 2 plus square root of straight x squared plus 4 straight x plus 1 end root close square brackets therefore space dy over dx equals fraction numerator 1 over denominator straight x plus 2 plus square root of straight x squared plus 4 straight x plus 1 end root end fraction cross times straight d over dx open square brackets straight x plus 2 plus square root of straight x squared plus 4 straight x plus 1 end root close square brackets equals fraction numerator 1 over denominator straight x plus 2 plus square root of straight x squared plus 4 straight x plus 1 end root end fraction cross times open square brackets 1 plus 0 plus fraction numerator 2 straight x plus 4 over denominator 2 square root of straight x squared plus 4 straight x plus 1 end root end fraction close square brackets open square brackets because straight d over dx left parenthesis straight x squared plus 4 straight x plus 1 right parenthesis to the power of begin inline style 1 half end style end exponent equals 1 half left parenthesis straight x squared plus 4 straight x plus 1 right parenthesis to the power of negative begin inline style 1 half end style end exponent. straight d over dx left parenthesis straight x squared plus 4 straight x plus 1 right parenthesis equals fraction numerator 2 straight x plus 4 over denominator 2 square root of straight x squared plus 4 straight x plus 1 end root end fraction close square brackets equals fraction numerator 1 over denominator straight x plus 2 plus square root of straight x squared plus 4 straight x plus 1 end root end fraction cross times open square brackets 1 plus fraction numerator straight x plus 2 over denominator square root of straight x squared plus 4 straight x plus 1 end root end fraction close square brackets equals fraction numerator 1 over denominator straight x plus 2 plus square root of straight x squared plus 4 straight x plus 1 end root end fraction cross times fraction numerator square root of straight x squared plus 4 straight x plus 1 end root plus straight x plus 2 over denominator square root of straight x squared plus 4 straight x plus 1 end root end fraction equals fraction numerator 1 over denominator square root of straight x squared plus 4 straight x plus 1 end root end fraction

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