Answer
The equation of the tangent line is $y=-3x+2$.
Work Step by Step
The slope $m$ of the tangent line of the curve $f(x)$ at $A(a,b)$ is also the slope of the curve at that point, which is calculated by $$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
$$f(x)=x-2x^2\hspace{1cm}A(1,-1)$$
1) Find the slope $m$ of the tangent:
$$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
Here $a=1$ and $f(a)=b=-1$
$$m=\lim_{h\to0}\frac{f(1+h)-(-1)}{h}=\lim_{h\to0}\frac{(1+h)-2(1+h)^2+1}{h}$$ $$m=\lim_{h\to0}\frac{1+h-2(1+2h+h^2)+1}{h}$$ $$m=\lim_{h\to0}\frac{1+h-2-4h-2h^2+1}{h}$$ $$m=\lim_{h\to0}\frac{-2h^2-3h}{h}=\lim_{h\to0}(-2h-3)$$ $$m=-2\times0-3=-3$$
2) Find the equation of the tangent line at $A(1,-1)$:
The tangent line would have this form: $$y=-3x+m$$
Substitute $A(1,-1)$ here to find $m$:
$$-3\times1+m=-1$$ $$-3+m=-1$$ $$m=2$$
So the equation of the tangent line is $y=-3x+2$.
