Is this do-able without an $x$ in the equation?

Question

I need help

Let's say you get a curve: $y=\frac{x-2}{2x+1}$

And the points are $(-1,3)$

You need to find the equation of the tangent

I differentiated it and it's derivative is $\frac{1}{2}$

Problem is, I can no longer sub in $x$ because there is no more $x$ in the derivative!

Did I do something wrong or is there another step to proceed further?

Answer 1

The derivative of this function is $$y'(x)=\frac{1}{2 x+1}-\frac{2 (x-2)}{(2 x+1)^2}=\frac{5}{(2 x+1)^2}$$ which can be found by using the product rule $$\frac{d}{dx}f(x)g(x)=f'(x)g(x)+g'(x)f(x)$$ using $f(x)=x-2$ and $g(x)=\left(2x+1\right)^{-1}$. Plugging in $x=-1$ gives $y'(-1)=5$. This is the slope of your tangent. You can find the equation of your tangent using $y=mx+b$ with $m=5$ and $b$ can be found by using your given point $(x=-1,y=3)$ together with this equation.

Answer 2

Actuelly your derivative is given by $$y'=\frac{2x+1-(x-2)2}{(2x+1)^2}=\frac{5}{(2x+1)^2}$$ so $$y'(-1)=5$$ and so do will get $$y=5x+n$$ to compute $n$ plug in this equation $x=-1,y=3$