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Differentiate the following w.r.t.x: (x2+2)4x2+5 - Mathematics and Statistics

Sum

Differentiate the following w.r.t.x: `(x^2 + 2)^4/(sqrt(x^2 + 5)`

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Solution

Let y = `(x^2 + 2)^4/(sqrt(x^2 + 5)`
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"[(x^2 + 2)^4/(sqrt(x^2 + 5))]`

= `(sqrt(x^2 + 5)."d"/"dx"(x^2 + 2)^4 - (x^2 + 2)^4."d"/"dx"(sqrt(x^2 + 5)))/(sqrt(x^2 + 5))^2`

= `(sqrt(x^2 + 5) xx 4(x^2 + 2)^3."d"/"dx"(x^2 + 2) - (x^2 + 2)^4 xx 1/(2(sqrt(x^2 + 5)))."d"/"dx"(x^2 + 5))/(x^2 + 5)`

= `(sqrt(x^2 + 5) xx 4(x^2 + 2)^3.(2x + 0) - (x^2 + 2)^4/(2sqrt(x^2 + 5)) xx (2x + 0))/(x^2 + 5)`

= `(8x(x^2 + 5)(x^2 + 2)^3 - x(x^2 + 2)^4)/(x^2 + 5)^(3/2)`

= `(x(x^2 + 2)^3[8(x^2 + 5) - (x^2 + 2)])/(x^2 + 5)^(3/2)`

= `(x(x^2 + 2)^3(8x^2 + 40 - x^2 - 2))/(x^2 + 5)^(3/2)`

= `(x(x^2 + 2)^3(7x^2 + 38))/(x^2 + 5)^(3/2)`.

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