What is the tangent of (2x+1)/(x+2),\at a=1 ?

The tangent of (2x+1)/(x+2),\at a=1 is y= 3/((a+2)^2)x+(2a^2+2a+2)/((a+2)^2)

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tangent of (2x+1)/(x+2),\at a=1

tangent of

\frac{2 x + 1}{x + 2}, at a = 1

y = \frac{3}{( a + 2 ) ^{2}} x + \frac{2 a ^{2} + 2 a + 2}{( a + 2 ) ^{2}}

Solution steps

Compute the tangent line to the general point

x = a

Find the tangent point:

(a, \frac{2 a + 1}{a + 2})

Find the slope of

y = \frac{2 x + 1}{x + 2} : \frac{d y}{d x} = \frac{3}{( x + 2 ) ^{2}}

Compute the slope of

y = \frac{2 x + 1}{x + 2}

x = a : m = \frac{3}{( a + 2 ) ^{2}}

Find the line with slope m=

\frac{3}{( a + 2 ) ^{2}}

and passing through

(a, \frac{2 a + 1}{a + 2}) : y = \frac{3}{( a + 2 ) ^{2}} x + \frac{2 a ^{2} + 2 a + 2}{( a + 2 ) ^{2}}

y = \frac{3}{( a + 2 ) ^{2}} x + \frac{2 a ^{2} + 2 a + 2}{( a + 2 ) ^{2}}

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What is the tangent of (2x+1)/(x+2),\at a=1 ?
The tangent of (2x+1)/(x+2),\at a=1 is y= 3/((a+2)^2)x+(2a^2+2a+2)/((a+2)^2)