Question

Calcule as derivadas da imagem

Answer 1

$\Large\boxed{\underline{\sf{Derivada~da~func_{\!\!,}\tilde{a}o~exponencial}}}\\\boxed{\boxed{\boxed{\boxed{\sf{\dfrac{d}{dx}(a^u)=a^u\cdot\ell n(a)\cdot\dfrac{du}{dx}}}}}}$

$\Large\boxed{\underline{\sf{Derivada~da~func_{\!\!,}\tilde{a}o~logar\acute{i}tmica}}}\\\boxed{\boxed{\boxed{\boxed{\sf{\dfrac{d}{dx}(\ell og_au)=\dfrac{1}{u\cdot\ell n(a)}\cdot\dfrac{du}{dx}}}}}}$

$\Large\boxed{\underline{\sf{Derivada~da~func_{\!\!,}\tilde{a}o~seno}}}\\\boxed{\boxed{\boxed{\boxed{\sf{\dfrac{d}{dx}(sen(u))=cos(u)\cdot\dfrac{du}{dx}}}}}}$

$\Large\boxed{\underline{\sf{derivada~da~pot\hat{e}ncia}}}\\\boxed{\boxed{\boxed{\boxed{\sf{\dfrac{d}{dx}(u^n)=nu^{n-1}\cdot\dfrac{du}{dx}}}}}}$

$\dotfill$

$\tt{a)}~\sf{y=e^xsen(x)\ell n(x)+\sqrt[5]{x^2}}\\\Large\sf{y=\left[e^xsen(x)\right]\ell n(x)+x^{\frac{2}{5}}}\\\sf{\dfrac{dy}{dx}=\left[e^xsen(x)+e^xcos(x)\right]\cdot\ell n(x)+\left[e^xsen(x)\right]\cdot\dfrac{1}{x}+\dfrac{2}{5}x^{-\frac{3}{5}}}$

$\tt{c)}~\sf{y=\dfrac{cossec^{-1}(x)}{cotg^{-1}(x)}}\\\sf{\dfrac{dy}{dx}=\dfrac{\frac{-1}{x\sqrt{x^2-1}}\cdot cotg^{-1}(x)-cossec^{-1}(x)\cdot\frac{-1}{1+x^2}}{(cotg^{-1}(x))^2}}\\\Large\boxed{\sf{\dfrac{dy}{dx}=\dfrac{-\frac{cotg^{-1}(x)}{x\sqrt{x^2-1}}+\frac{cossec^{-1}(x)}{1+x^2}}{\left[cotg^{-1}(x)\right]^2}}}$