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

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tangent of x^{sin(x)}

tangent of

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}})

f (x) = \frac{3 x}{( x + 1 ) ^{2}}, at x = 0

f (x) = \frac{1}{x ^{2} + 1}

\frac{d}{d x} (4 sin^{3} (x))

\frac{d}{d x} ((lo g_{10} (x))^{l o g_{10} (x)})

y^{^{'}} = 2 - y