Küpkök x'in Türevi

Küpkök x'in Türevi Nedir ?

Küpkök x'in türevi 1/(3.∛x²)'tir.

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

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

Küpkök x'in Türevinin İspatı

1. Yol

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

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

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

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

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

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

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

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

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

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

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

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

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

2. Yol

$y=\sqrt[3]{x}$

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

$y^3=x$

$(y^3{)}^{\prime }=(x{)}^{\prime }$

$\boxed{(u^n{)}^{\prime }=n.(u{)}^{n-1}.u^{\prime }}$

$3.y^2.y^{\prime }=1$

$y^{\prime }=\frac{1}{3.y^2}$

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

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

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

3. Yol

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

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

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

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

$\boxed{(lnu{)}^{\prime }=\frac{u^{\prime }}{u}}$

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

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

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

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

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