1/x'in türevi

1/x'in türevi nedir ?

$f(x)=\frac{1}{x}\Rightarrow f^{\prime }(x)=-\frac{1}{x^2}$

1/x'in türevinin ispatı

1. Yol

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

$(\frac{1}{x}{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{1}{x+h}-\frac{1}{x}}{h}$

$(\frac{1}{x}{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{x-(x+h)}{x.(x+h)}}{h}$

$(\frac{1}{x}{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{\cancel{x}-\cancel{x}-h}{x^2+x.h}}{h}$

$(\frac{1}{x}{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{-\cancel{h}}{x^2+x.h}}{\cancel{h}}$

$(\frac{1}{x}{)}^{\prime }=\underset{h\to 0}{\lim }-\frac{1}{x^2+x.h}$

$(\frac{1}{x}{)}^{\prime }=-\frac{1}{x^2+x.0}$

$(\frac{1}{x}{)}^{\prime }=-\frac{1}{x^2+0}$

$(\frac{1}{x}{)}^{\prime }=-\frac{1}{x^2}$

2. Yol

$f(x)=\frac{1}{x}$

$f(x)=x^{-1}$

$lnf(x)=ln{x}^{-1}$

$lnf(x)=-1.lnx$

$lnf(x)=-lnx$

$[lnf(x){]}^{\prime }=(-lnx{)}^{\prime }$

$[lnf(x){]}^{\prime }=-(lnx{)}^{\prime }$

$\boxed{f(x)=lng(x)\Rightarrow f^{\prime }(x)=\frac{g^{\prime }(x)}{g(x)}}$

$\frac{f^{\prime }(x)}{f(x)}=-\frac{(x{)}^{\prime }}{x}$

$\frac{f^{\prime }(x)}{f(x)}=-\frac{1}{x}$

$f^{\prime }(x)=-\frac{1}{x}.f(x)$

$f^{\prime }(x)=-\frac{1}{x}.\frac{1}{x}$

$f^{\prime }(x)=-\frac{1.1}{x.x}$

$f^{\prime }(x)=-\frac{1}{x^2}$

3. Yol

$f(x)=\frac{1}{x}$

$f(x).x=1$

$[f(x).x]=(1{)}^{\prime }$

$\boxed{f(x)=u(x).v(x)\Rightarrow f^{\prime }(x)=u^{\prime }(x).v(x)+v^{\prime }(x).u(x)}$

$f^{\prime }(x).x+(x{)}^{\prime }.f(x)=0$

$f^{\prime }(x).x+1.f(x)=0$

$f^{\prime }(x).x+f(x)=0$

$f^{\prime }(x).x=-f(x)$

$f^{\prime }(x)=-\frac{f(x)}{x}$

$f^{\prime }(x)=-\frac{\frac{1}{x}}{x}$

$f^{\prime }(x)=-\frac{\frac{1}{x}}{\frac{x}{1}}$

$f^{\prime }(x)=-\frac{1}{x}.\frac{1}{x}$

$f^{\prime }(x)=-\frac{1.1}{x.x}$

$f^{\prime }(x)=-\frac{1}{x^2}$