What is the tangent of y=(x^3-4x)^{10} ?

The tangent of y=(x^3-4x)^{10} is

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tangent of y=(x^3-4x)^{10}

tangent of

y = (x^{3} - 4 x)^{10}

y = 10 (a_{0}^{3} - 4 a_{0})^{9} (3 a_{0}^{2} - 4) x - 29 a_{0}^{30} + 1080 a_{0}^{28} - 18000 a_{0}^{26} + 176640 a_{0}^{24} - 1128960 a_{0}^{22} + 4902912 a_{0}^{20} - 14622720 a_{0}^{18} + 29491200 a_{0}^{16} - 38338560 a_{0}^{14} + 28835840 a_{0}^{12} - 9437184 a_{0}^{10}

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})^{10})

Find the slope of

y = (x^{3} - 4 x)^{10} : \frac{d y}{d x} = 10 (x^{3} - 4 x)^{9} (3 x^{2} - 4)

Compute the slope of

y = (x^{3} - 4 x)^{10}

x = a_{0} : m = 10 (a_{0}^{3} - 4 a_{0})^{9} (3 a_{0}^{2} - 4)

Find the line with slope m=

10 (a_{0}^{3} - 4 a_{0})^{9} (3 a_{0}^{2} - 4)

and passing through

(a_{0}, (a_{0}^{3} - 4 a_{0})^{10}) : y = 10 (a_{0}^{3} - 4 a_{0})^{9} (3 a_{0}^{2} - 4) x - 29 a_{0}^{30} + 1080 a_{0}^{28} - 18000 a_{0}^{26} + 176640 a_{0}^{24} - 1128960 a_{0}^{22} + 4902912 a_{0}^{20} - 14622720 a_{0}^{18} + 29491200 a_{0}^{16} - 38338560 a_{0}^{14} + 28835840 a_{0}^{12} - 9437184 a_{0}^{10}

y = 10 (a_{0}^{3} - 4 a_{0})^{9} (3 a_{0}^{2} - 4) x - 29 a_{0}^{30} + 1080 a_{0}^{28} - 18000 a_{0}^{26} + 176640 a_{0}^{24} - 1128960 a_{0}^{22} + 4902912 a_{0}^{20} - 14622720 a_{0}^{18} + 29491200 a_{0}^{16} - 38338560 a_{0}^{14} + 28835840 a_{0}^{12} - 9437184 a_{0}^{10}

\frac{d}{d x} (\frac{5}{2} x)

\frac{3}{sin ( x ) + cos ( x )}

y^{^{'}} = x + y, y (0) = 5

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

x \to - \infty lim (x - 1)