How many tangent lines to the curve y=\frac{x}{x+1} pass through

Joan Thompson 2021-12-13 Answered
How many tangent lines to the curve y=xx+1 pass through the point (1,2)
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Expert Answer

Ana Robertson
Answered 2021-12-14 Author has 26 answers
The form of the tangent lines must be:
y=m(x1)+2 [1]
Now, substitute the first derivative of the curve for m
dydx=d(x)dx(x+1)xd(x+1)dx(x+1)2
dydx=x+1x(x+1)2
dydx=1(x+1)2
m=1(x+1)2 [2]
Now, substitute equation [2] into equation [1]
y=x1(x+1)2+2 [1.1]
We have to substitute y=xx+1
xx+1=x1(x+1)2+2
Solve for x
x(x+1)=(x1)+2(x+1)2
x2+x=x1+2x2+4x+2
x2+4x+1
x=4±424(1)(1)2(1)
x=2±3
x=2+3 and x=23
There are two tangent lines
y=1(1+3)2(x1)+2
and
y=1(13)2(x1)+2
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