How do you find the derivatives of $s=roott(t)$ by logarithmic differentiation?

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Answer 1 · 2017-03-15T19:06:16

Please see the explanation.

Given: $s = root(t)(t)$

Rewrite as:

$s=t^(1/t)$

Use the natural logarithm on both sides:

$ln(s)=ln(t^(1/t))$

Use the property $ln(a^c) = (c)ln(a)$

$ln(s)=(1/t)ln(t)$

Differentiate both sides:

The left side is trivial but we need the quotient rule, $(g/h)'=(g'h-gh')/h^2$ for the right side:

$g=ln(t)$
$h= t$
$g'=1/t$
$h'=1$

$(1/t)ln(t) = ((1/t)(t)-ln(t)(1))/t^2=(1-ln(t))/t^2$

$1/s(ds)/(dt) = (1-ln(t))/t^2$

Multiply both sides by s:

$(ds)/(dt) = (s(1-ln(t)))/t^2$

Substitute for s:

$(ds)/(dt) = (root(t)(t)(1-ln(t)))/t^2$

How do you find the derivatives of s=roott(t)s=roott(t) by logarithmic differentiation?