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Question
Differentiate the following with respect to x.
`1/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 = `1/sqrt(1 + x^2)`
y = `(1 + x^2)^(- 1/2)`
Here n = `- 1/2`; f(x) = 1 + x2
`"dy"/"dx" = - 1/2(1 + x^2)^(- 1/2 - 1) "d"/"dx" (1 + x^2)`
`= - 1/2 (1 + x^2)^(- 3/2)`(0 + 2x)
`= - 1/2 1/(1 + x^2)^(3/2)`(2x)
`= - x/(sqrt (1 + x^2))^3`
`= (-x)/(sqrt((x + x^2)^2) sqrt(1 + x^2))`
`= (-x)/((1 + x^2)sqrt(1 + x^2))`
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