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Question
Differentiate the following with respect to x.
`sqrt(1 + x^2)`
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Solution
For the following problems chain rule to be used:
`"d"/"dx"` f(g(x)) = f'(g(x)) . g'(x)
`"d"/"dx"` [f(x)]n = n[f(x)]n-1 × `"d"/"dx"`f(x)
Let y = `sqrt(1 + x^2)`
y = `(1 + x^2)^(1/2)`
Here f(x) = 1 + x2; n = `1/2`
`= 1/2 (1 + x^2)^(1/2 - 1) "d"/"dx" (1 + x^2)`
`= 1/2 (1 + x^2)^(-1/2) (0 + 2x)`
`= 1/2 1/(1 + x^2)^(1/2) (2x)`
`= 1/2 1/sqrt (1 + x^2) (2x)`
`= x/(sqrt (1 + x^2)`
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