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Question
Differentiate the following w.r.t.x:
`sqrt(x^2 + sqrt(x^2 + 1)`
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Solution
Let y = `sqrt(x^2 + sqrt(x^2 + 1)`
Differentiating w.r.t. x, we get,
`"dy"/"dx" = "d"/"dx"(x^2 + sqrt(x^2 + 1))^(1/2)`
= `(1)/(2)(x^2 + sqrt(x^2 + 1))^(-1/2)."d"/"dx"(x^2 + sqrt(x^2 + 1))`
= `(1)/(2sqrt(x^2 + sqrt(x^2 + 1))).["d"/"dx"(x^2) + "d"/"dx"(sqrt(x^2 + 1))]`
= `(1)/(2sqrt(x^2 + sqrt(x^2 + 1))).[2x + 1/(2sqrt(x^2 + 1)). "d"/"dx"(x^2 + 1)]`
= `(1)/(2sqrt(x^2 + sqrt(x^2 + 1))).[2x + 1/(2sqrt(x^2 + 1)) 2x]`
= `(1)/(2sqrt(x^2 + sqrt(x^2 + 1))).[2x + x/(sqrt(x^2 + 1))]`
= `(1)/(2sqrt(x^2 + sqrt(x^2 + 1))).[(2xsqrt(x^2+1) + x)/(sqrt(x^2 + 1))]`
= `(x (2sqrt(x^2 + 1) + 1))/(2sqrt(x^2 +1).sqrt(x^2 + sqrt(x^2 + 1))`
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