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)

Compute the slope of

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

x = a_{0} : 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}

\int x cos^{2} (x) d x

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

\frac{d}{d x} (sec (2 x) tan (2 x))

\frac{\partial}{\partial x} (x - z - arctan (yz))

f (x) = \frac{x ^{3}}{( x ^{2} + 4 ) ^{2}}