Question

What is the equation of the tangent line of `f(x)=(2x+1)/(x+2)` at `x=1`?

Answer 1

`y-1=(1/3)(x-1)`

Explanation:

You are given the x value so plug that into the original equation to find y: `(2(1)+1)/((1)+2)` and you get `1`, now we have a point `(1,1)` so we now know we are looking for the slope of the tangent line at `(1,1)`

The tangent line is the derivative of the equation so find the derivative using the quotient rule:
`(g(x)f'(x)-f(x)g'(x))/(g(x)^2)` where `g(x)` is the denominator and `f(x)` is the numerator so you get:

`f'(x)=3/(x+2)^2` as the derivative
Now plug in 1 for the x value and `f'(x)` will return you with the slope or `m`. You now get `1/3` for `f'(x)` this is you `m` value now place everything in equation form:

`y-1 = 1/3(x-1)`
solve for y if necessary