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Question
Differentiate the following w.r.t. x:
`sqrt(e^(sqrtx))`, x > 0
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Solution
Let, y = `sqrt(e^(sqrtx))`
Differentiating both sides with respect to x,
`dy/dx = d/dx (e^(sqrtx))^(1/2)`
= `1/2 (e^(sqrtx))^(1/2 - 1) d/dx e^(sqrtx)`
= `1/2 (e^(sqrtx))^(-1/2) e^(sqrtx) d/dx x^(1/2)`
= `1/2 (e^(sqrtx))^(-1/2) e^(sqrtx) 1/2 x^(-1/2)`
= `1/4 1/(sqrt(e^(sqrtx))) e^(sqrtx) 1/sqrtx`
= `1/4 e^(sqrtx)/(sqrt(x. e^(sqrtx)))`, x > 0
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