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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $log open parentheses square root of straight x minus straight a end root plus square root of straight x minus straight b end root close parentheses$

Solution

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

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