What is the tangent of y=x^{sin(x)} ?

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tangent of y=x^{sin(x)}

tangent of

y = x^{s i n (x)}

y = a_{0}^{s i n (a_{0})} (cos (a_{0}) ln (a_{0}) + \frac{sin ( a _{0} )}{a _{0}}) x + a_{0}^{s i n (a_{0})} - a_{0}^{s i n (a_{0}) + 1} (ln (a_{0}) cos (a_{0}) + \frac{sin ( a _{0} )}{a _{0}})

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, a_{0}^{s i n (a_{0})})

Find the slope of

y = x^{s i n (x)} : \frac{d y}{d x} = x^{s i n (x)} (cos (x) ln (x) + \frac{sin ( x )}{x})

Compute the slope of

y = x^{s i n (x)}

x = a_{0} : m = a_{0}^{s i n (a_{0})} (cos (a_{0}) ln (a_{0}) + \frac{sin ( a _{0} )}{a _{0}})

Find the line with slope m=

a_{0}^{s i n (a_{0})} (cos (a_{0}) ln (a_{0}) + \frac{sin ( a _{0} )}{a _{0}})

and passing through

(a_{0}, a_{0}^{s i n (a_{0})}) : y = a_{0}^{s i n (a_{0})} (cos (a_{0}) ln (a_{0}) + \frac{sin ( a _{0} )}{a _{0}}) x + a_{0}^{s i n (a_{0})} - a_{0}^{s i n (a_{0}) + 1} (ln (a_{0}) cos (a_{0}) + \frac{sin ( a _{0} )}{a _{0}})

y = a_{0}^{s i n (a_{0})} (cos (a_{0}) ln (a_{0}) + \frac{sin ( a _{0} )}{a _{0}}) x + a_{0}^{s i n (a_{0})} - a_{0}^{s i n (a_{0}) + 1} (ln (a_{0}) cos (a_{0}) + \frac{sin ( a _{0} )}{a _{0}})

t^{2} - 8 t + 16

\frac{d}{d x} ((2 x)^{6 x})

x \to - \infty lim (x^{2} e^{- 3 x})

x - e^{x}

\int (x - 1)^{- 2} d x