What is the tangent of (x^2+1)^{(x+1)} ?

The tangent of (x^2+1)^{(x+1)} is

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tangent of (x^2+1)^{(x+1)}

tangent of

(x^{2} + 1)^{(x + 1)}

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

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, (a_{0}^{2} + 1)^{a_{0} + 1})

Find the slope of

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

Compute the slope of

y = (x^{2} + 1)^{x + 1}

x = a_{0} : m = (ln (a_{0}^{2} + 1) + \frac{2 a _{0} ( a _{0} + 1 )}{a _{0}^{2} + 1}) (a_{0}^{2} + 1)^{a_{0} + 1}

Find the line with slope m=

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

and passing through

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

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

\frac{d}{d x} (\frac{2 x + 5}{3})

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x \to 0 lim (\frac{1}{x ^{\frac{2}{3}}})

\frac{d}{d s} (ln (s^{2} + 1))

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