Question

Qual a derivada da raiz quadrada de 1-x^4?

Answer 1

$\textbf{Regra da cadeia:}\\\\ \mathrm{y=h(g(x))}\ \to\ \boxed{\mathrm{\dfrac{dy}{dx}=h'(g(x)).g'(x)}}\\\\ \textbf{Resolu\c{c}\~ao:}\\\\ \mathrm{y=\sqrt{1-x^4}\ \to\ \dfrac{dy}{dx}=\big[(1-x^4)^{\frac{1}{2}}\big]'\big[1-x^4\big]'=}\\\\ \mathrm{=\dfrac{1}{2}(1-x^4)^{\frac{1}{2}-1}(0-4x^3)=\dfrac{1}{2}(1-x^4)^{-\frac{1}{2}}(-4x^3)=}\\\\ \mathrm{=-\dfrac{2x^3}{(1-x^4)^{\frac{1}{2}}}=\boxed{\mathbf{-\dfrac{2x^3}{\sqrt{1-x^4}}}}}$