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Question
Differentiate the following:
y = `sqrt(x +sqrt(x)`
Solution
y = `sqrt(x +sqrt(x)`
y = `[x + x^(1/2)]^(1/2)`
y = f(g(x))
`("d"y)/("d"x)` = f'(g(x)) . g'(x)
`("d"y)/("d"x) = 1/2 [x + x^(1/2)]^(1/2 - 1) [1 + 1/2 x^(1/2 - 1)]`
`("d"y)/("d"x) = 1/2[x + x^(1/2)]^(- 1/2) [1 + 1/2 x^(- 1/2)]`
= `1/[2x + x^(1/2)]^(1/2) [1 + 1/(2x^(1/2))]`
= `1/(2sqrt(x + sqrt(x))) xx [(2x^(1/2) + 1)/(2x^(1/2))]`
= `1/(2sqrt(x + sqrt(x))) xx [(2sqrt(x) + 1)/(2sqrt(x))]`
`("d"y)/("d"x) = (2sqrt(x) + 1)/(4sqrt(x) * sqrt(x sqrt(x))`
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