Question

derivada de raiz de (a² - x²)

Answer 1

Resposta:

Obs.: como não dito nada, estarei derivando em relação a x, assim considerando ''a'' uma constante.

Pela regra da cadeia, na qual

$\boxed{\sf \dfrac{d}{dx}[f(u)]=\dfrac{d}{du}(f(u))\cdot \dfrac{d}{dx}(u)}$

, faça u = a² - x²:

$\sf\dfrac{d}{dx}\big(\sqrt{a^2-x^2}\big)=\dfrac{d}{du}(\sqrt{u})\cdot \dfrac{d}{dx}(u)$

$\sf\dfrac{d}{dx}\big(\sqrt{a^2-x^2}\big)=\dfrac{d}{du}(u^{\frac{1}{2}})\cdot\dfrac{d}{dx}(u)$

$\sf\dfrac{d}{dx}\big(\sqrt{a^2-x^2}\big)=\dfrac{1}{2}\cdot u^{\frac{1}{2}-1}\cdot \dfrac{d}{dx}(u)$

$\sf \dfrac{d}{dx}\big(\sqrt{a^2-x^2}\big)=\dfrac{1}{2}\cdot u^{-\frac{1}{2}}\cdot \dfrac{d}{dx}(u)$

$\sf \dfrac{d}{dx}\big(\sqrt{a^2-x^2}\big)=\dfrac{1}{2}\cdot \dfrac{1}{u^{\frac{1}{2}}}\cdot \dfrac{d}{dx}(u)$

$\sf \dfrac{d}{dx}\big(\sqrt{a^2-x^2}\big)=\dfrac{\frac{d}{dx}(u)}{2u^{\frac{1}{2}}}$

$\sf \dfrac{d}{dx}\big(\sqrt{a^2-x^2}\big)=\dfrac{\frac{d}{dx}(a^2-x^2)}{2\sqrt{a^2-x^2}}$

$\sf \dfrac{d}{dx}\big(\sqrt{a^2-x^2}\big)=\dfrac{\frac{d}{dx}(a^2)-\frac{d}{dx}(x^2)}{2\sqrt{a^2-x^2}}$

$\sf \dfrac{d}{dx}\big(\sqrt{a^2-x^2}\big)=\dfrac{0-2x}{2\sqrt{a^2-x^2}}$

$\sf \dfrac{d}{dx}\big(\sqrt{a^2-x^2}\big)=\dfrac{-\,2x}{2\sqrt{a^2-x^2}}$

$\red{\sf \dfrac{d}{dx}\big(\sqrt{a^2-x^2}\big)=-\dfrac{x}{\sqrt{a^2-x^2}}}$