What is the tangent of (x^2+1)/(x+e^x+2) ?

The tangent of (x^2+1)/(x+e^x+2) is

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tangent of (x^2+1)/(x+e^x+2)

tangent of

\frac{x ^{2} + 1}{x + e ^{x} + 2}

y = \frac{a _{0}^{2} - e ^{a_{0}} a _{0}^{2} + 2 e ^{a_{0}} a _{0} + 4 a _{0} - e ^{a_{0}} - 1}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}} x + \frac{a _{0}^{3} e ^{a_{0}} - a _{0}^{2} e ^{a_{0}} - 2 a _{0}^{2} + 2 a _{0} + a _{0} e ^{a_{0}} + e ^{a_{0}} + 2}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, \frac{a _{0}^{2} + 1}{a _{0} + e ^{a_{0}} + 2})

Find the slope of

y = \frac{x ^{2} + 1}{x + e ^{x} + 2} : \frac{d y}{d x} = \frac{x ^{2} - e ^{x} x ^{2} + 2 e ^{x} x + 4 x - e ^{x} - 1}{( x + e ^{x} + 2 ) ^{2}}

EN : T i tl e G e n er a l Eq u a t i o n Sl o p e A t P o in t 2 Eq : m = \frac{a _{0}^{2} - e ^{a_{0}} a _{0}^{2} + 2 e ^{a_{0}} a _{0} + 4 a _{0} - e ^{a_{0}} - 1}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}}

Find the line with slope m=

\frac{a _{0}^{2} - e ^{a_{0}} a _{0}^{2} + 2 e ^{a_{0}} a _{0} + 4 a _{0} - e ^{a_{0}} - 1}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}}

and passing through

(a_{0}, \frac{a _{0}^{2} + 1}{a _{0} + e ^{a_{0}} + 2}) : y = \frac{a _{0}^{2} - e ^{a_{0}} a _{0}^{2} + 2 e ^{a_{0}} a _{0} + 4 a _{0} - e ^{a_{0}} - 1}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}} x + \frac{a _{0}^{3} e ^{a_{0}} - a _{0}^{2} e ^{a_{0}} - 2 a _{0}^{2} + 2 a _{0} + a _{0} e ^{a_{0}} + e ^{a_{0}} + 2}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}}

y = \frac{a _{0}^{2} - e ^{a_{0}} a _{0}^{2} + 2 e ^{a_{0}} a _{0} + 4 a _{0} - e ^{a_{0}} - 1}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}} x + \frac{a _{0}^{3} e ^{a_{0}} - a _{0}^{2} e ^{a_{0}} - 2 a _{0}^{2} + 2 a _{0} + a _{0} e ^{a_{0}} + e ^{a_{0}} + 2}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}}

\int (- x^{2} - 3 x) d x

W (A) = \frac{A}{6}

f (x) = \frac{4 x ^{3}}{( 3 - 2 x ) ^{5}}

x \to 1 lim (x^{2} - 5)

y^{^{'}} = y + x