What is the tangent of (x^3-9x)^{10}(3) ?

The tangent of (x^3-9x)^{10}(3) is

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tangent of (x^3-9x)^{10}(3)

tangent of

(x^{3} - 9 x)^{10} (3)

y = 30 (a_{0}^{3} - 9 a_{0})^{9} (3 a_{0}^{2} - 9) x - 87 a_{0}^{30} + 7290 a_{0}^{28} - 273375 a_{0}^{26} + 6036120 a_{0}^{24} - 86802030 a_{0}^{22} + 848179836 a_{0}^{20} - 5691733110 a_{0}^{18} + 25828032600 a_{0}^{16} - 75546995355 a_{0}^{14} + 127848761370 a_{0}^{12} + 9^{10} \cdot 3 a_{0}^{10} - 104603532030 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} - 9 a_{0})^{10} \cdot 3)

Find the slope of

y = (x^{3} - 9 x)^{10} 3 : \frac{d y}{d x} = 30 (x^{3} - 9 x)^{9} (3 x^{2} - 9)

Compute the slope of

y = (x^{3} - 9 x)^{10} 3

x = a_{0} : m = 30 (a_{0}^{3} - 9 a_{0})^{9} (3 a_{0}^{2} - 9)

Find the line with slope m=

30 (a_{0}^{3} - 9 a_{0})^{9} (3 a_{0}^{2} - 9)

and passing through

(a_{0}, (a_{0}^{3} - 9 a_{0})^{10} 3) : y = 30 (a_{0}^{3} - 9 a_{0})^{9} (3 a_{0}^{2} - 9) x - 87 a_{0}^{30} + 7290 a_{0}^{28} - 273375 a_{0}^{26} + 6036120 a_{0}^{24} - 86802030 a_{0}^{22} + 848179836 a_{0}^{20} - 5691733110 a_{0}^{18} + 25828032600 a_{0}^{16} - 75546995355 a_{0}^{14} + 127848761370 a_{0}^{12} + 9^{10} \cdot 3 a_{0}^{10} - 104603532030 a_{0}^{10}

y = 30 (a_{0}^{3} - 9 a_{0})^{9} (3 a_{0}^{2} - 9) x - 87 a_{0}^{30} + 7290 a_{0}^{28} - 273375 a_{0}^{26} + 6036120 a_{0}^{24} - 86802030 a_{0}^{22} + 848179836 a_{0}^{20} - 5691733110 a_{0}^{18} + 25828032600 a_{0}^{16} - 75546995355 a_{0}^{14} + 127848761370 a_{0}^{12} + 9^{10} \cdot 3 a_{0}^{10} - 104603532030 a_{0}^{10}

(\frac{2 x}{x ^{2} + 1})^{^{'}}

x \to 4 lim (4 x^{2} - 5 x + 1)

ln (x + 3)

2 x^{2} + 7

y^{^{'}} - 2 y = - 1, y (0) = 3