What is the tangent of x^{1/3+y^{1/3}}=7 ?

The tangent of x^{1/3+y^{1/3}}=7 is

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tangent of x^{1/3+y^{1/3}}=7

tangent of

x^{\frac{1}{3} + y^{\frac{1}{3}}} = 7

y = (\frac{2 ln ^{2} ( 7 )}{a _{0} ln ^{3} ( a _{0} )} - \frac{3 ln ^{3} ( 7 )}{a _{0} ln ^{4} ( a _{0} )} - \frac{ln ( 7 )}{3 a _{0} ln ^{2} ( a _{0} )}) x + \frac{ln ^{3} ( 7 ) - 2 ln ^{2} ( 7 )}{ln ^{3} ( a _{0} )} - \frac{ln ^{2} ( 7 )}{ln ^{2} ( a _{0} )} + \frac{ln ( 7 )}{3 ln ( a _{0} )} - \frac{1}{27} + \frac{3 ln ^{3} ( 7 )}{ln ^{4} ( a _{0} )} + \frac{ln ( 7 )}{3 ln ^{2} ( a _{0} )}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, \frac{ln ^{3} ( 7 )}{ln ^{3} ( a _{0} )} - \frac{ln ^{2} ( 7 )}{ln ^{2} ( a _{0} )} + \frac{ln ( 7 )}{3 ln ( a _{0} )} - \frac{1}{27})

Find the slope of

x^{\frac{1}{3} + y^{\frac{1}{3}}} = 7 : \frac{d y}{d x} = \frac{2 ln ^{2} ( 7 )}{x ln ^{3} ( x )} - \frac{3 ln ^{3} ( 7 )}{x ln ^{4} ( x )} - \frac{ln ( 7 )}{3 x ln ^{2} ( x )}

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{2 ln ^{2} ( 7 )}{a _{0} ln ^{3} ( a _{0} )} - \frac{3 ln ^{3} ( 7 )}{a _{0} ln ^{4} ( a _{0} )} - \frac{ln ( 7 )}{3 a _{0} ln ^{2} ( a _{0} )}

Find the line with slope m=

\frac{2 ln ^{2} ( 7 )}{a _{0} ln ^{3} ( a _{0} )} - \frac{3 ln ^{3} ( 7 )}{a _{0} ln ^{4} ( a _{0} )} - \frac{ln ( 7 )}{3 a _{0} ln ^{2} ( a _{0} )}

and passing through

(a_{0}, \frac{ln ^{3} ( 7 )}{ln ^{3} ( a _{0} )} - \frac{ln ^{2} ( 7 )}{ln ^{2} ( a _{0} )} + \frac{ln ( 7 )}{3 ln ( a _{0} )} - \frac{1}{27}) : y = (\frac{2 ln ^{2} ( 7 )}{a _{0} ln ^{3} ( a _{0} )} - \frac{3 ln ^{3} ( 7 )}{a _{0} ln ^{4} ( a _{0} )} - \frac{ln ( 7 )}{3 a _{0} ln ^{2} ( a _{0} )}) x + \frac{ln ^{3} ( 7 ) - 2 ln ^{2} ( 7 )}{ln ^{3} ( a _{0} )} - \frac{ln ^{2} ( 7 )}{ln ^{2} ( a _{0} )} + \frac{ln ( 7 )}{3 ln ( a _{0} )} - \frac{1}{27} + \frac{3 ln ^{3} ( 7 )}{ln ^{4} ( a _{0} )} + \frac{ln ( 7 )}{3 ln ^{2} ( a _{0} )}

y = (\frac{2 ln ^{2} ( 7 )}{a _{0} ln ^{3} ( a _{0} )} - \frac{3 ln ^{3} ( 7 )}{a _{0} ln ^{4} ( a _{0} )} - \frac{ln ( 7 )}{3 a _{0} ln ^{2} ( a _{0} )}) x + \frac{ln ^{3} ( 7 ) - 2 ln ^{2} ( 7 )}{ln ^{3} ( a _{0} )} - \frac{ln ^{2} ( 7 )}{ln ^{2} ( a _{0} )} + \frac{ln ( 7 )}{3 ln ( a _{0} )} - \frac{1}{27} + \frac{3 ln ^{3} ( 7 )}{ln ^{4} ( a _{0} )} + \frac{ln ( 7 )}{3 ln ^{2} ( a _{0} )}