Find the equation of the tangent line to the graph of $g(x)=e^{x^7-8x}$ at $(-1,e^7)$
$Solution:$
The value of the slope of the tangent line to $g(x)$ at the point $(-1,e^7)$ is $g'(-1)$, so lets calculate that.
$g'(x)=e^{x^7-8x}(x^7-8x)'=e^{x^7-8x}(7x^6-8)$
$\rightarrow g'(-1)=e^{(-1)^7-8(-1)}(7(-1)^6-8)=e^{7}(-1)=-e^7$
So we know that $y=-e^7x+b$. Let's plug in the given point to solve for $b$.
$e^7=-e^7(-1)+b$
$\rightarrow b=0$
Thus our equation is $y=-e^7x$
Well done! Your equation is correct.