Question

determine a derivada da função de f(X)=e₢os x

Answer 1

Regra da Cadeia. Se $u$ e $v$ são funções de $x$, então

$\dfrac{d}{dx}\left[u\circ v\left(x \right ) \right ]=\dfrac{d}{dx}\left[u\left(v\left(x \right ) \right ) \right ]\\ \\ =\dfrac{du}{dv}\cdot \dfrac{dv}{dx}$


$f\left(x \right )=e^{\cos x}$

$\bullet\;\;f\left(x \right )=u\left(v \right )=e^{v}\\ \\ \bullet\;\;v\left(x \right )=\cos x$


$\dfrac{df}{dx}=\dfrac{d}{dx}\left(e^{\cos x} \right )\\ \\ \dfrac{df}{dx}=\dfrac{d}{dx}\left(e^{u} \right )\\ \\ \dfrac{df}{dx}=\dfrac{d}{dv}\left(e^{v} \right )\cdot \dfrac{dv}{dx}\\ \\ \dfrac{df}{dx}=e^{v}\cdot\dfrac{d}{dx}\left(\cos x \right ) \\ \\ \dfrac{df}{dx}=e^{\cos x}\cdot\left(-\mathrm{sen\, }x \right )\\ \\ \boxed{\dfrac{df}{dx}=-e^{\cos x}\cdot \mathrm{\,sen\, }x}$