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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $square root of straight e to the power of square root of straight x end exponent end root comma straight X greater than 0$

Solution

Let space space space space space space space square root of straight e to the power of square root of straight x end exponent end root equals open parentheses straight e to the power of square root of straight x end exponent close parentheses to the power of 1 half end exponent therefore space space space space space dy over dx equals 1 half open parentheses straight e to the power of square root of straight x end exponent close parentheses to the power of 1 half end exponent. straight d over dx open parentheses straight e to the power of square root of straight x end exponent close parentheses equals fraction numerator 1 over denominator 2 square root of straight e to the power of square root of straight x end exponent end root end fraction. straight e to the power of square root of straight x end exponent. straight d over dx open parentheses square root of straight x close parentheses space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 square root of straight e to the power of square root of straight x end exponent end root end fraction. straight e to the power of square root of straight x end exponent. fraction numerator 1 over denominator 2 square root of straight x end fraction equals 1 fourth fraction numerator straight e to the power of square root of straight x end exponent over denominator square root of straight x space straight e to the power of square root of straight x end exponent end root end fraction

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