What is the tangent of ((x^4+1))/(e^x+2x+1) ?

The tangent of ((x^4+1))/(e^x+2x+1) is

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tangent of ((x^4+1))/(e^x+2x+1)

tangent of

\frac{( x ^{4} + 1 )}{e ^{x} + 2 x + 1}

y = \frac{6 a _{0}^{4} - e ^{a_{0}} a _{0}^{4} + 4 e ^{a_{0}} a _{0}^{3} + 4 a _{0}^{3} - e ^{a_{0}} - 2}{( e ^{a_{0}} + 2 a _{0} + 1 ) ^{2}} x + \frac{- 4 a _{0}^{5} + a _{0}^{5} e ^{a_{0}} - 3 a _{0}^{4} e ^{a_{0}} - 3 a _{0}^{4} + 4 a _{0} + a _{0} e ^{a_{0}} + e ^{a_{0}} + 1}{( 2 a _{0} + e ^{a_{0}} + 1 ) ^{2}}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

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

Find the slope of

y = \frac{x ^{4} + 1}{e ^{x} + 2 x + 1} : \frac{d y}{d x} = \frac{6 x ^{4} - e ^{x} x ^{4} + 4 e ^{x} x ^{3} + 4 x ^{3} - e ^{x} - 2}{( e ^{x} + 2 x + 1 ) ^{2}}

Compute the slope of

y = \frac{x ^{4} + 1}{e ^{x} + 2 x + 1}

x = a_{0} : m = \frac{6 a _{0}^{4} - e ^{a_{0}} a _{0}^{4} + 4 e ^{a_{0}} a _{0}^{3} + 4 a _{0}^{3} - e ^{a_{0}} - 2}{( e ^{a_{0}} + 2 a _{0} + 1 ) ^{2}}

Find the line with slope m=

\frac{6 a _{0}^{4} - e ^{a_{0}} a _{0}^{4} + 4 e ^{a_{0}} a _{0}^{3} + 4 a _{0}^{3} - e ^{a_{0}} - 2}{( e ^{a_{0}} + 2 a _{0} + 1 ) ^{2}}

and passing through

(a_{0}, \frac{a _{0}^{4} + 1}{e ^{a_{0}} + 2 a _{0} + 1}) : y = \frac{6 a _{0}^{4} - e ^{a_{0}} a _{0}^{4} + 4 e ^{a_{0}} a _{0}^{3} + 4 a _{0}^{3} - e ^{a_{0}} - 2}{( e ^{a_{0}} + 2 a _{0} + 1 ) ^{2}} x + \frac{- 4 a _{0}^{5} + a _{0}^{5} e ^{a_{0}} - 3 a _{0}^{4} e ^{a_{0}} - 3 a _{0}^{4} + 4 a _{0} + a _{0} e ^{a_{0}} + e ^{a_{0}} + 1}{( 2 a _{0} + e ^{a_{0}} + 1 ) ^{2}}

y = \frac{6 a _{0}^{4} - e ^{a_{0}} a _{0}^{4} + 4 e ^{a_{0}} a _{0}^{3} + 4 a _{0}^{3} - e ^{a_{0}} - 2}{( e ^{a_{0}} + 2 a _{0} + 1 ) ^{2}} x + \frac{- 4 a _{0}^{5} + a _{0}^{5} e ^{a_{0}} - 3 a _{0}^{4} e ^{a_{0}} - 3 a _{0}^{4} + 4 a _{0} + a _{0} e ^{a_{0}} + e ^{a_{0}} + 1}{( 2 a _{0} + e ^{a_{0}} + 1 ) ^{2}}

12 x^{3} - 24 x^{2}

f (I) = I^{22}

\int \frac{x ^{2} + 8}{x ^{4} - x ^{3} - 2 x ^{2}} d x

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

f (x) = \frac{7}{( x ^{2} - 10 ) ^{4}}

What is the tangent of ((x^4+1))/(e^x+2x+1) ?
The tangent of ((x^4+1))/(e^x+2x+1) is