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Question
Differentiate the following:
f(t) = `root(3)(1 + tan "t")`
Solution
f(t) = `root(3)(1 + tan "t")`
f(t) = `(1 + tan "t")^(1/3)`
[y = f(g(x)
`("d"y)/("d"x)` = f'(g(x)) . g'(x)]
`f"'"("t") = 1/3 (1 + tan "t")^(1/3 - 1) "d"/"dt" (1 + tan "t")`
= `1/3 (1 + tan "t")^(- 2/3) (0 + sec^2 "t")`
= `1/3 xx 1/(1 + tan "t")^(2/3) xx sec^2"t"`
= `(sec^2"t")/(3(1 + tan "t")^(2/3)`
`f"'"("t") = 1/2 sec^2"t" (1 + tan "t")^(- 2/3)`
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