Question

derivada de f(x) Raiz de (x-3)

Answer 1

Regra da cadeia:

$\boxed{\boxed{\dfrac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)=\dfrac{dy}{du}\dfrac{du}{dx}}}$

Lembrando da derivada de √x:

$y=\sqrt{x}=x^{(1/2)}~~~~~\therefore~~~~~\boxed{\boxed{y'=\dfrac{1}{2\sqrt{x}}}}$
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Seja g(x) = √x e h(x) = x - 3, então, a composta de g e h é dada por:

$g(x)=\sqrt{x}\\g(h(x))=\sqrt{h(x)}\\g(h(x))=\sqrt{x-3}\\\\\boxed{\boxed{f(x)=g(h(x))=\sqrt{x-3}}}$

Acharemos a derivada de f(x) pela regra da cadeia:

$\dfrac{d}{dx}f(x)=\dfrac{1}{2\sqrt{x-3}}\cdot\dfrac{d}{dx}(x-3)\\\\\\f'(x)=\dfrac{1}{2\sqrt{x-3}}\cdot(1\cdot x^{1-1}+0})\\\\\\\boxed{\boxed{f'(x)=\dfrac{1}{2\sqrt{x-3}}}}$