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Question
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))]`
`dy/dx = (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`
`dy/dx = (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)`
`dy/dx = (sqrt (x^2 + 5) xx 4(x^2 + 2)^3 * (2x + 0) - (x^2 + 2)^4/(2 sqrt (x^2 + 5)) xx (2x + 0))/(x^2 + 5)`
`dy/dx = (8x (x^2 + 5) (x^2 + 2)^3 - x (x^2 + 2)^4)/(x^2 + 5)^(3/2)`
`dy/dx = (x (x^2 + 2)^3 [8 (x^2 + 5) - (x^2 + 2)])/(x^2 + 5)^(3/2)`
`dy/dx = (x (x^2 + 2)^3 (8x^2 + 40 - x^2 - 2))/(x^2 + 5)^(3/2)`
`dy/dx = (x (x^2 + 2)^3 (7x^2 + 38))/(x^2 + 5)^(3/2)`.
Notes
Question is modified as per the answer given in the textbook.
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