Karekök x'in Türevi

Karekök x'in Türevi Nedir ?

Karekök x'in türevi 1/(2.√x)'tir.

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

$\frac{d}{dx}(\sqrt{x})=\frac{1}{2.\sqrt{x}}$

Karekök x'in Türevinin İspatı

1. Yol

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

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

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

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

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

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

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

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

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

$(\sqrt{x}{)}^{\prime }=\frac{1}{\sqrt{x}+\sqrt{x}}$

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

2. Yol

$y=\sqrt{x}$

$y^2=(\sqrt{x}{)}^2$

$y^2=x$

$(y^2{)}^{\prime }=(x{)}^{\prime }$

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

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

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

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

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

3. Yol

$\sqrt{x}=x^{\frac{1}{2}}$

$ln\sqrt{x}=ln{x}^{\frac{1}{2}}$

$ln\sqrt{x}=\frac{1}{2}.lnx$

$(ln\sqrt{x}{)}^{\prime }=(\frac{1}{2}.lnx{)}^{\prime }$

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

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

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

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

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

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