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Question
Differentiate the following:
s(t) = `root(4)(("t"^3 + 1)/("t"^3 - 1)`
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Solution
s(t) = `root(4)(("t"^3 + 1)/("t"^3 - 1)`
= `(("t"^3 + 1)/("t"^3 - 1))^(1/4)`
s'(t) = `1/4(("t"^3 + 1)/("t"^3 - 1))^(1/4 - 1) xx "d"/("d"x) (("t"^3 + 1)/("t"^3 - 1))`
= `1/4 (("t"^3 + 1)/("t"^3 - 1))^(- 3/4) xx (("t"^3 - 1)(3"t"^2 + 0) - ("t"^2 + 0) - ("t"^3 + 1)(3"t"^2 - 0))/("t"^3 - 1)^2`
= `1/4 (("t"^3 + 1)/("t"^3 - 1))^(- 3/4) xx (3"t"^5 - 3"t"^2 - 3"t"^5 - 3"t"^2)/("t"^3 - 1)^2`
= `1/4 (("t"^3 + 1)/("t"^3 - 1))^(- 3/4) xx (- 6"t"^2)/("t"^3 - 1)^2`
= `(- 3"t"^2 xx ("t"^3 + 1)^(- 3/4))/(2("t"^3 - 1)^(- 3/4) ("t"^3 - 1)^2`
= `(- 3"t"^2 xx ("t"^3 + 1)^(- 3/4))/(2("t"^3 - 1)^(- 3/4 + 2)`
= `(- 3"t"^2)/(2("t"^3 + 1)^(3/4) ("t"^3 - 1)^((- 3 + 8)/4)`
s'(t) = `(- 3"t"^2)/(2("t"^3 + 1)^(3/4) ("t"^3 - 1)^(5/4))`
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