What is the tangent of (4x)/(x+2) ?

The tangent of (4x)/(x+2) is y= 8/((a_{0)+2)^2}x+(4a_{0}^2)/((a_{0)+2)^2}

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tangent of (4x)/(x+2)

tangent of

\frac{4 x}{x + 2}

y = \frac{8}{( a _{0} + 2 ) ^{2}} x + \frac{4 a _{0}^{2}}{( a _{0} + 2 ) ^{2}}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, \frac{4 a _{0}}{a _{0} + 2})

Find the slope of

y = \frac{4 x}{x + 2} : \frac{d y}{d x} = \frac{8}{( x + 2 ) ^{2}}

Compute the slope of

y = \frac{4 x}{x + 2}

x = a_{0} : m = \frac{8}{( a _{0} + 2 ) ^{2}}

Find the line with slope m=

\frac{8}{( a _{0} + 2 ) ^{2}}

and passing through

(a_{0}, \frac{4 a _{0}}{a _{0} + 2}) : y = \frac{8}{( a _{0} + 2 ) ^{2}} x + \frac{4 a _{0}^{2}}{( a _{0} + 2 ) ^{2}}

y = \frac{8}{( a _{0} + 2 ) ^{2}} x + \frac{4 a _{0}^{2}}{( a _{0} + 2 ) ^{2}}

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What is the tangent of (4x)/(x+2) ?
The tangent of (4x)/(x+2) is y= 8/((a_{0)+2)^2}x+(4a_{0}^2)/((a_{0)+2)^2}