What is the tangent of (x^3-25x)^{12} ?

The tangent of (x^3-25x)^{12} is

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tangent of (x^3-25x)^{12}

tangent of

(x^{3} - 25 x)^{12}

y = 12 (a_{0}^{3} - 25 a_{0})^{11} (3 a_{0}^{2} - 25) x - 35 a_{0}^{36} + 9900 a_{0}^{34} - 1278750 a_{0}^{32} + 99687500 a_{0}^{30} - 5220703125 a_{0}^{28} + 193359375000 a_{0}^{26} - 5188476562500 a_{0}^{24} + 2 5^{7} \cdot 11088 a_{0}^{22} + 3.38379 E 13 a_{0}^{22} - 2 5^{8} \cdot 5445 a_{0}^{20} - 2 5^{7} \cdot 99000 a_{0}^{20} + 2 5^{9} \cdot 1760 a_{0}^{18} + 2 5^{8} \cdot 49500 a_{0}^{18} - 2 5^{10} \cdot 330 a_{0}^{16} - 2 5^{9} \cdot 16500 a_{0}^{16} + 2 5^{11} \cdot 24 a_{0}^{14} + 2 5^{10} \cdot 3300 a_{0}^{14} - 2 5^{12} \cdot 11 a_{0}^{12}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, (a_{0}^{3} - 25 a_{0})^{12})

Find the slope of

y = (x^{3} - 25 x)^{12} : \frac{d y}{d x} = 12 (x^{3} - 25 x)^{11} (3 x^{2} - 25)

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 = 12 (a_{0}^{3} - 25 a_{0})^{11} (3 a_{0}^{2} - 25)

Find the line with slope m=

12 (a_{0}^{3} - 25 a_{0})^{11} (3 a_{0}^{2} - 25)

and passing through

(a_{0}, (a_{0}^{3} - 25 a_{0})^{12}) : y = 12 (a_{0}^{3} - 25 a_{0})^{11} (3 a_{0}^{2} - 25) x - 35 a_{0}^{36} + 9900 a_{0}^{34} - 1278750 a_{0}^{32} + 99687500 a_{0}^{30} - 5220703125 a_{0}^{28} + 193359375000 a_{0}^{26} - 5188476562500 a_{0}^{24} + 2 5^{7} \cdot 11088 a_{0}^{22} + 3.38379 E 13 a_{0}^{22} - 2 5^{8} \cdot 5445 a_{0}^{20} - 2 5^{7} \cdot 99000 a_{0}^{20} + 2 5^{9} \cdot 1760 a_{0}^{18} + 2 5^{8} \cdot 49500 a_{0}^{18} - 2 5^{10} \cdot 330 a_{0}^{16} - 2 5^{9} \cdot 16500 a_{0}^{16} + 2 5^{11} \cdot 24 a_{0}^{14} + 2 5^{10} \cdot 3300 a_{0}^{14} - 2 5^{12} \cdot 11 a_{0}^{12}

y = 12 (a_{0}^{3} - 25 a_{0})^{11} (3 a_{0}^{2} - 25) x - 35 a_{0}^{36} + 9900 a_{0}^{34} - 1278750 a_{0}^{32} + 99687500 a_{0}^{30} - 5220703125 a_{0}^{28} + 193359375000 a_{0}^{26} - 5188476562500 a_{0}^{24} + 2 5^{7} \cdot 11088 a_{0}^{22} + 3.38379 E 13 a_{0}^{22} - 2 5^{8} \cdot 5445 a_{0}^{20} - 2 5^{7} \cdot 99000 a_{0}^{20} + 2 5^{9} \cdot 1760 a_{0}^{18} + 2 5^{8} \cdot 49500 a_{0}^{18} - 2 5^{10} \cdot 330 a_{0}^{16} - 2 5^{9} \cdot 16500 a_{0}^{16} + 2 5^{11} \cdot 24 a_{0}^{14} + 2 5^{10} \cdot 3300 a_{0}^{14} - 2 5^{12} \cdot 11 a_{0}^{12}

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