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Question
If y = `root(5)((3"x"^2 + 8"x" + 5)^4)`, find `"dy"/"dx"`.
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Solution
y = `root(5)((3"x"^2 + 8"x" + 5)^4)`
∴ y = `(3"x"^2 + 8"x" + 5)^(4/5)`
Differentiating both sides w.r.t. x, we get
`"dy"/"dx" = "d"/"dx" [(3"x"^2 + 8"x" + 5)^(4/5)]`
`= 4/5(3"x"^2 + 8"x" + 5)^(-1/5) * "d"/"dx" (3"x"^2 + 8"x" + 5)`
`= 4/5(3"x"^2 + 8"x" + 5)^(-1/5) * [3(2"x") + 8 + 0]`
∴ `"dy"/"dx" = 4/5(3"x"^2 + 8"x" + 5)^(-1/5) * (6"x" + 8)`
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