küpkök x'in türevi

küpkök x'in türevi nedir ?

$f(x)=\sqrt[3]{x}\Rightarrow f^{\prime }(x)=\frac{1}{3\sqrt[3]{x^2}}$

küpkök x'in türevinin ispatı

1. Yol

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

$(\sqrt[3]{x}{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\sqrt[3]{x+h}-\sqrt[3]{x}}{h}$

$(\sqrt[3]{x}{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\sqrt[3]{x+h}-\sqrt[3]{x}}{h}.\frac{(\sqrt[3]{x+h}{)}^2+\sqrt[3]{x+h}.\sqrt[3]{x}+(\sqrt[3]{x}{)}^2}{(\sqrt[3]{x+h}{)}^2+\sqrt[3]{x+h}.\sqrt[3]{x}+(\sqrt[3]{x}{)}^2}$

$(\sqrt[3]{x}{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\sqrt[3]{x+h}-\sqrt[3]{x}}{h}.\frac{\sqrt[3]{(x+h{)}^2}+\sqrt[3]{(x+h).x}+\sqrt[3]{x^2}}{\sqrt[3]{(x+h{)}^2}+\sqrt[3]{(x+h).x}+\sqrt[3]{x^2}}$

$(\sqrt[3]{x}{)}^{\prime }=\underset{h\to 0}{\lim }\frac{(\sqrt[3]{x+h}-\sqrt[3]{x}).(\sqrt[3]{(x+h{)}^2}+\sqrt[3]{(x+h).x}+\sqrt[3]{x^2})}{h.(\sqrt[3]{(x+h{)}^2}+\sqrt[3]{(x+h).x}+\sqrt[3]{x^2})}$

$\boxed{a^3-b^3=(a-b).(a^2+ab+b^2)}$

$(\sqrt[3]{x}{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\sqrt[3]{(x+h{)}^3}-\sqrt[3]{x^3}}{h.(\sqrt[3]{(x+h{)}^2}+\sqrt[3]{(x+h).x}+\sqrt[3]{x^2})}$

$(\sqrt[3]{x}{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\cancel{x}+h-\cancel{x}}{h.(\sqrt[3]{x^2+2.x.h+h^2}+\sqrt[3]{x^2+x.h}+\sqrt[3]{x^2})}$

$(\sqrt[3]{x}{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\cancel{h}}{\cancel{h}.(\sqrt[3]{x^2+2.x.h+h^2}+\sqrt[3]{x^2+x.h}+\sqrt[3]{x^2})}$

$(\sqrt[3]{x}{)}^{\prime }=\underset{h\to 0}{\lim }\frac{1}{\sqrt[3]{x^2+2.x.h+h^2}+\sqrt[3]{x^2+x.h}+\sqrt[3]{x^2}}$

$(\sqrt[3]{x}{)}^{\prime }=\frac{1}{\sqrt[3]{x^2+2.x.0+0^2}+\sqrt[3]{x^2+x.0}+\sqrt[3]{x^2}}$

$(\sqrt[3]{x}{)}^{\prime }=\frac{1}{\sqrt[3]{x^2+0+0}+\sqrt[3]{x^2+0}+\sqrt[3]{x^2}}$

$(\sqrt[3]{x}{)}^{\prime }=\frac{1}{\sqrt[3]{x^2}+\sqrt[3]{x^2}+\sqrt[3]{x^2}}$

$(\sqrt[3]{x}{)}^{\prime }=\frac{1}{3.\sqrt[3]{x^2}}$

2. Yol

$f(x)=\sqrt[3]{x}$

$[f(x){]}^3=(\sqrt[3]{x}{)}^3$

$[f(x){]}^3=\sqrt[3]{x^3}$

$[f(x){]}^3=x$

$\{[f(x){]}^3{\}}^{\prime }=(x{)}^{\prime }$

$\boxed{f(x)=[g(x){]}^n\Rightarrow f^{\prime }(x)=n.[g(x){]}^{n-1}.g^{\prime }(x)}$

$3.[f(x){]}^{3-1}.f^{\prime }(x)=1$

$3.[f(x){]}^2.f^{\prime }(x)=1$

$f^{\prime }(x)=\frac{1}{3.[f(x){]}^2}$

$f^{\prime }(x)=\frac{1}{3.(\sqrt[3]{x}{)}^2}$

$f^{\prime }(x)=\frac{1}{3.\sqrt[3]{x^2}}$

3. Yol

$\sqrt[3]{x}=x^{\frac{1}{3}}$

$ln\sqrt[3]{x}=ln{x}^{\frac{1}{3}}$

$ln\sqrt[3]{x}=\frac{1}{3}.lnx$

$(ln\sqrt[3]{x}{)}^{\prime }=(\frac{1}{3}.lnx{)}^{\prime }$

$(ln\sqrt[3]{x}{)}^{\prime }=\frac{1}{3}.(lnx{)}^{\prime }$

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

$\frac{(\sqrt[3]{x}{)}^{\prime }}{\sqrt[3]{x}}=\frac{1}{3}.\frac{(x{)}^{\prime }}{x}$

$\frac{(\sqrt[3]{x}{)}^{\prime }}{\sqrt[3]{x}}=\frac{1}{3}.\frac{1}{x}$

$\frac{(\sqrt[3]{x}{)}^{\prime }}{\sqrt[3]{x}}=\frac{1.1}{3.x}$

$\frac{(\sqrt[3]{x}{)}^{\prime }}{\sqrt[3]{x}}=\frac{1}{3.x}$

$(\sqrt[3]{x}{)}^{\prime }=\sqrt[3]{x}.\frac{1}{3x}$

$(\sqrt[3]{x}{)}^{\prime }=\frac{\sqrt[3]{x}.1}{3.x}$

$(\sqrt[3]{x}{)}^{\prime }=\frac{\sqrt[3]{x}}{3.x}$

$(\sqrt[3]{x}{)}^{\prime }=\frac{\sqrt[3]{x}}{3.\sqrt[3]{x^3}}$

$(\sqrt[3]{x}{)}^{\prime }=\frac{\sqrt[3]{x}}{3.\sqrt[3]{x^2.x}}$

$(\sqrt[3]{x}{)}^{\prime }=\frac{\cancel{\sqrt[3]{x}}}{3.\sqrt[3]{x^2}.\cancel{\sqrt[3]{x}}}$

$(\sqrt[3]{x}{)}^{\prime }=\frac{1}{3.\sqrt[3]{x^2}}$