Question

Derivada de e^x/x^2. Como resolvo?

Answer 1

Regra do quociente:

$\boxed{\boxed{\dfrac{d}{dx}\left[\dfrac{f(x)}{g(x)}\right]=\left[\dfrac{f(x)}{g(x)}\right]'=\dfrac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^{2}}}}$
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$y=\dfrac{e^{x}}{x^{2}}$

Se tomarmos f(x) = e^x e g(x) = x², temos

$y=\dfrac{e^{x}}{x^{2}}=\dfrac{f(x)}{g(x)}$

Achando as derivadas de f e g:

$f'(x)=\dfrac{d}{dx}(e^{x})=e^{x}\\\\\\g'(x)=\dfrac{d}{dx}(x^{2})=2x$

Aplicando a regra do quociente para achar a derivada de y:

$y'=\dfrac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^{2}}\\\\\\y'=\dfrac{e^{x}\cdot x^{2}-e^{x}\cdot2x}{(x^{2})^{2}}$

Colocando xe^x em evidência no numerador:

$y'=\dfrac{x\cdot e^{x}\cdot(x-2)}{x^{4}}\\\\\\\boxed{\boxed{y'=\dfrac{e^{x}\cdot(x-2)}{x^{3}}}}$