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Differentiate the following w.r.t.x: x2+x2+1

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Question

Differentiate the following w.r.t.x:

`sqrt(x^2 + sqrt(x^2 + 1)`

Sum
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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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Chapter 1: Differentiation - Exercise 1.1 [Page 11]

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