Derivative of Exponential

Por partes,

$e^{\ln y} = \alpha \\\\ \log_e \alpha = \ln y \\\\ \ln \alpha = \ln y \\\\ \boxed{\alpha = y }$

Isto posto,

$y = 3 - x^2 \cdot e^{\ln y} \\\\ y = 3 - x^2 \cdot y \\\\y + x^2y = 3 \\\\ y(1 + x^2) = 3 \\\\ y = \frac{3}{1 + x^2} \\\\ f(x) = \frac{3}{1 + x^2} \\\\ f'(x) = \frac{(3)' \cdot (1 + x^2) - (3) \cdot (1 + x^2)'}{(1 + x^2)^2} \\\\ f'(x) = \frac{- 3 \cdot 2x}{(1 + x^2)^2} \\\\ f'(- 2) = \frac{- 3 \cdot 2 \cdot (- 2)}{(5)^2} \\\\ \boxed{f'(- 2) = \frac{12}{25}}$

$y=3-x^{2}e^{lny}$

$y=3-x^{2}y$

$y+x^{2}y=3$

$y(1+x^{2})=3$

$y=\frac{3}{1+x^{2}}$

$f(x)=y=\frac{3}{1+x^{2}}$

$f'(x)=y'=\frac{-6x}{(1+x^{2})^2}$

$f'(-2)=\frac{-6(-2)}{(1+(-2)^{2})^2}$

$f'(-2)=\frac{12}{(1+4)^2}$

$f'(-2)=\frac{12}{(5)^2}$

$f'(-2)=\boxed{\frac{12}{25}}$