Answer
The equation of the tangent line is $y=\frac{12x-16}{25}.$
Work Step by Step
Using the quotient rule: $fâ(x)=(\frac{u(x)}{v(x)})'=\frac{u'(x)v(x)-v'(x)u(x)}{(v(x))^2}$
$u(x)=16x; u'(x)=16$
$v(x)=x^2+16; v'(x)=2x$
$f'(x)=\frac{(16)(x^2+16)-(16x)(2x)}{(x^2+16)^2}=-\frac{16(x^2-16)}{(x^2+16)^2}$
$f'(-2)=-\frac{16((-2)^2-16)}{((-2)^2+16)^2}=\frac{12}{25}$.
Equation of tangent:
$(y-y_0)=m(x-x_0)$ at point $(x_0, y_0)$ and slope $m$.
$(y+\frac{8}{5})=\frac{12}{25}(x+2)\rightarrow y=\frac{12x-16}{25}.$
