4.4 Logaritmifunktion derivaatta

Lause

$D\ln x=\frac{1}{x}{,}\ kun\ x>0$

$D\ln\left|x\right|=\frac{1}{x}{,}\ kun\ x\ne0$

Esim. Derivoi, kun x>0

$D2\ln x^3=D2\cdot3\ln x=6D\ln x=6\cdot\frac{1}{x}=\frac{6}{x}$

b)
$\ln\ \frac{7x}{3}$
$D\ln\ \frac{7x}{3}=D\ln\ \frac{7}{3}x=\frac{1}{\frac{7x}{3}}\cdot\frac{7}{3}=\left(\frac{7x}{3}\right)^{-1}\cdot\frac{7}{3}=\frac{3}{7x}\cdot\frac{7}{3}=\frac{1}{x}$

$\ln\left(3\sqrt[]{x}\right){,}\ u\left(x\right)=\ln x{,}\ u'\left(x\right)=\frac{1}{x}{,}\ s\left(x\right)=3\sqrt[]{x}=x^{\frac{1}{3}}{,}\ s'\left(x\right)=\frac{1}{3}x^{-\frac{2}{3}}$

$D\ln\left(\sqrt[3]{x}\right)=\frac{1}{x^{\frac{1}{3}}}\cdot\frac{1}{3}x^{-\frac{2}{3}}=x^{-\frac{1}{3}}\cdot\frac{1}{3}x^{-\frac{2}{3}}=\frac{1}{3}\cdot\left(x^{^{-\frac{1}{3}+\left(-\frac{2}{3}\right)}}\right)=\frac{1}{3}\cdot x^{-1}=\frac{1}{3x}$

482.

$f\left(x\right)=\ln\left(x^3-x\right)$

$x^3-x>0{,}\ kun\ -1<x<0\ tai\ x>1\ \left(laskin\right)$

$f'\left(x\right)=\frac{1}{x^3-x}\cdot\left(3x^2-1\right)=\frac{3x^2-1}{x^3-x}$

Ratkaistaan derivaattafunktion nollakohdat

$x=\frac{-\sqrt[]{3}}{3}\ tai\ x=\frac{\sqrt[]{x}}{3}$
$x=-0{,}57735\ tai\ x=0{,}57735$

Ratkaisuista toteuttaa määrittelyehdon vain

$x=-\frac{\sqrt[]{3}}{3}\approx-0{,}577$
Laaditaan funktion f(x) kulkukaavio, Selvitetään kulkukaaviota varten derivaatan merkit testikohtien avulal. Nimetään f'(x)=g(x)

$g\left(2\right)=\frac{11}{6}$

$g\left(-0{,}1\right)=-9{,}79798$
$g\left(-0{,}7\right)=1{,}31653$

$\begin{array}{l|l} &-1&&-\frac{\sqrt[]{3}}{3}&&0&&1&\\ \hline f'\left(x\right)&&+&&-&&\setminus&&+\\ f\left(x\right)&&\nearrow&&\searrow&&\setminus&&\nearrow \end{array}$

Funktiolla on paikallinen maksimiarvo kohdassa

$x=-\frac{\sqrt[]{3}}{3}$

$f\left(-\frac{\sqrt[]{3}}{3}\right)=\frac{\ln\ \frac{4}{27}}{2}$

Sievennetään laskimen antaman tulos

$\frac{\ln\ \frac{4}{27}}{2}=\frac{1}{2}\cdot\ln\ \left(\frac{4}{27}\right)=\ln\ \left(\frac{4}{27}\right)^{\frac{1}{2}}=\sqrt[]{\frac{4}{27}}=\ln\ \frac{\sqrt[]{4}}{\sqrt[]{27}}=\ln\ \frac{2}{\sqrt[]{3\cdot9}}=\ln\ \frac{2}{3\sqrt[]{3}}$

Lause

$D\log_ax=\frac{1}{\ln a}\cdot\frac{1}{x}{,}\ kun\ x>0$