The document discusses the derivation of the logarithmic derivative of several functions: 1) For the function y = xln(x), the logarithmic derivative is y'/y = 1/x. 2) For the function y = 2^3x, the logarithmic derivative is y'/y = 3/2x. 3) For the function y = x(x^2-1)/(x^2+1)^(1/2), the logarithmic derivative is y'/y = (3x^2-1)/(x^3-x) - 4x/(x^2+1).

1. Derivacion logoritmica
3) 𝑦 = 𝑥ln 𝑥
, 𝑥 > 0
ln 𝑦 = ln 𝑥ln 𝑥
ln 𝑦 = ln 𝑥. ln 𝑥
ln 𝑦 = (ln 𝑥)2
𝑦´.
1
𝑦
= 2 𝑙𝑛 𝑥.
1
𝑥
𝑦´
𝑦
=.
2 ln 𝑥
𝑥
5) 𝑦 = 23𝑥
ln 𝑦 = ln 23𝑥
ln 𝑦 = 3𝑥 𝑙𝑛 (2)
𝑦´
𝑦
= 3.
1
2
𝑦´ =
3
2
𝑦
11) 𝑦 =
𝑥(𝑥2−1)
√𝑥2+1
𝑦 =
𝑥3−𝑥
(𝑥2+1)
1
2
⁄
𝑦 = (𝑥3
− 𝑥). (𝑥2
+ 1)
−1
2
⁄
ln 𝑦 = ln((𝑥3
− 𝑥)(𝑥2
+ 1)
−1
2
⁄
)
ln 𝑦 = (𝑥3
− 𝑥) + ln(𝑥2
+ 1)
−1
2
⁄
𝑦´.
1
𝑦
=
(3𝑥2−1)
(𝑥3−𝑥)
−
2
(𝑥2+1)
. (2𝑥)
𝑦´
𝑦
=
3𝑥2−1
(𝑥3−𝑥)
−
4𝑥
(𝑥2+1)
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