What is the tangent of (x^3-4x)^8 ?

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tangent of (x^3-4x)^8

tangent of

(x^{3} - 4 x)^{8}

y = 8 (a_{0}^{3} - 4 a_{0})^{7} (3 a_{0}^{2} - 4) x - 23 a_{0}^{24} + 672 a_{0}^{22} - 8512 a_{0}^{20} + 60928 a_{0}^{18} - 268800 a_{0}^{16} + 745472 a_{0}^{14} - 1261568 a_{0}^{12} + 1179648 a_{0}^{10} - 458752 a_{0}^{8}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, (a_{0}^{3} - 4 a_{0})^{8})

Find the slope of

y = (x^{3} - 4 x)^{8} : \frac{d y}{d x} = 8 (x^{3} - 4 x)^{7} (3 x^{2} - 4)

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 = 8 (a_{0}^{3} - 4 a_{0})^{7} (3 a_{0}^{2} - 4)

Find the line with slope m=

8 (a_{0}^{3} - 4 a_{0})^{7} (3 a_{0}^{2} - 4)

and passing through

(a_{0}, (a_{0}^{3} - 4 a_{0})^{8}) : y = 8 (a_{0}^{3} - 4 a_{0})^{7} (3 a_{0}^{2} - 4) x - 23 a_{0}^{24} + 672 a_{0}^{22} - 8512 a_{0}^{20} + 60928 a_{0}^{18} - 268800 a_{0}^{16} + 745472 a_{0}^{14} - 1261568 a_{0}^{12} + 1179648 a_{0}^{10} - 458752 a_{0}^{8}

y = 8 (a_{0}^{3} - 4 a_{0})^{7} (3 a_{0}^{2} - 4) x - 23 a_{0}^{24} + 672 a_{0}^{22} - 8512 a_{0}^{20} + 60928 a_{0}^{18} - 268800 a_{0}^{16} + 745472 a_{0}^{14} - 1261568 a_{0}^{12} + 1179648 a_{0}^{10} - 458752 a_{0}^{8}

\frac{\partial}{\partial x} (x^{2} + y^{2} - 1)

d er i v a t i v e f (x) = \frac{x ^{3}}{( x ^{2} + 4 ) ^{2}}

y^{^{'}} - \frac{3 y}{x} = x

d er i v a t i v e sin (x^{2}) + \frac{( 1 - x )}{1 + x}

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