[tex]\large\boxed{\begin{array}{l}\underline{\sf A\,derivada\,do\,logaritmo\,natural}\\\\\displaystyle\rm Como~\ln x=\int_1^x\dfrac{1}{t}~dt~para~x>0,segue\,do\\\rm teorema\,fundamental\,do\,c\acute alculo\,que\\\rm D_{x}\ln x=\dfrac{1}{x}. Em\,geral\,vale\,o\,seguinte\,teorema\\\underline{\sf A\,derivada\,do\,\ln u}\\\rm Se\,u\,\acute e\,uma\,func_{\!\!,}\tilde ao\,diferenci\acute avel\,de\,x\,e\,u>0,ent\tilde ao\\\rm D_{x}\ln u=\dfrac{1}{u}\cdot D_{x}u\end{array}}[/tex]
[tex]\large\boxed{\begin{array}{l}\underline{\bf Prova:}\\\rm J\acute a\,sabemos\,pela\,d~\!\!efinic_{\!\!,}\tilde ao\,de\,\ln\,e\,do\,teorema\\\rm fundamental\,do\,c\acute alculo\,que\,D_x\ln u=\dfrac{1}{u}.\\\rm Logo\,pela\,regra\,da\,cadeia\,D_x\ln u=\dfrac{1}{u}\cdot D_xu\end{array}}[/tex]
[tex]\large\boxed{\begin{array}{l}\rm Portanto\\\rm y(x)=\ell n(x^4)\\\rm D_xy(x)=\dfrac{1}{x^4}\cdot D_x(x^4)\\\\\rm D_xy(x)=\dfrac{1}{\diagup\!\!\!\!\!x^4}\cdot 4\diagup\!\!\!\!\!x^3\\\\\rm D_xy(x)=\dfrac{4}{x}\end{array}}[/tex]
