What is the tangent of (x^3-16x)^{14} ?

The tangent of (x^3-16x)^{14} is

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tangent of (x^3-16x)^{14}

tangent of

(x^{3} - 16 x)^{14}

y = 14 (a_{0}^{3} - 16 a_{0})^{13} (3 a_{0}^{2} - 16) x - 41 a_{0}^{42} + 8736 a_{0}^{40} - 861952 a_{0}^{38} + 52183040 a_{0}^{36} - 2164850688 a_{0}^{34} + 65076723712 a_{0}^{32} - 1461077409792 a_{0}^{30} + 2.48743 E 13 a_{0}^{28} - 1 6^{8} \cdot 51051 a_{0}^{26} - 1.03182 E 14 a_{0}^{26} + 1 6^{9} \cdot 28028 a_{0}^{24} + 1 6^{8} \cdot 288288 a_{0}^{24} - 1 6^{10} \cdot 11011 a_{0}^{22} - 1 6^{9} \cdot 160160 a_{0}^{22} + 1 6^{11} \cdot 2912 a_{0}^{20} + 1 6^{10} \cdot 64064 a_{0}^{20} - 1 6^{12} \cdot 455 a_{0}^{18} - 1 6^{11} \cdot 17472 a_{0}^{18} + 1 6^{13} \cdot 28 a_{0}^{16} + 1 6^{12} \cdot 2912 a_{0}^{16} - 1 6^{14} \cdot 13 a_{0}^{14}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, (a_{0}^{3} - 16 a_{0})^{14})

Find the slope of

y = (x^{3} - 16 x)^{14} : \frac{d y}{d x} = 14 (x^{3} - 16 x)^{13} (3 x^{2} - 16)

Compute the slope of

y = (x^{3} - 16 x)^{14}

x = a_{0} : m = 14 (a_{0}^{3} - 16 a_{0})^{13} (3 a_{0}^{2} - 16)

Find the line with slope m=

14 (a_{0}^{3} - 16 a_{0})^{13} (3 a_{0}^{2} - 16)

and passing through

(a_{0}, (a_{0}^{3} - 16 a_{0})^{14}) : y = 14 (a_{0}^{3} - 16 a_{0})^{13} (3 a_{0}^{2} - 16) x - 41 a_{0}^{42} + 8736 a_{0}^{40} - 861952 a_{0}^{38} + 52183040 a_{0}^{36} - 2164850688 a_{0}^{34} + 65076723712 a_{0}^{32} - 1461077409792 a_{0}^{30} + 2.48743 E 13 a_{0}^{28} - 1 6^{8} \cdot 51051 a_{0}^{26} - 1.03182 E 14 a_{0}^{26} + 1 6^{9} \cdot 28028 a_{0}^{24} + 1 6^{8} \cdot 288288 a_{0}^{24} - 1 6^{10} \cdot 11011 a_{0}^{22} - 1 6^{9} \cdot 160160 a_{0}^{22} + 1 6^{11} \cdot 2912 a_{0}^{20} + 1 6^{10} \cdot 64064 a_{0}^{20} - 1 6^{12} \cdot 455 a_{0}^{18} - 1 6^{11} \cdot 17472 a_{0}^{18} + 1 6^{13} \cdot 28 a_{0}^{16} + 1 6^{12} \cdot 2912 a_{0}^{16} - 1 6^{14} \cdot 13 a_{0}^{14}

y = 14 (a_{0}^{3} - 16 a_{0})^{13} (3 a_{0}^{2} - 16) x - 41 a_{0}^{42} + 8736 a_{0}^{40} - 861952 a_{0}^{38} + 52183040 a_{0}^{36} - 2164850688 a_{0}^{34} + 65076723712 a_{0}^{32} - 1461077409792 a_{0}^{30} + 2.48743 E 13 a_{0}^{28} - 1 6^{8} \cdot 51051 a_{0}^{26} - 1.03182 E 14 a_{0}^{26} + 1 6^{9} \cdot 28028 a_{0}^{24} + 1 6^{8} \cdot 288288 a_{0}^{24} - 1 6^{10} \cdot 11011 a_{0}^{22} - 1 6^{9} \cdot 160160 a_{0}^{22} + 1 6^{11} \cdot 2912 a_{0}^{20} + 1 6^{10} \cdot 64064 a_{0}^{20} - 1 6^{12} \cdot 455 a_{0}^{18} - 1 6^{11} \cdot 17472 a_{0}^{18} + 1 6^{13} \cdot 28 a_{0}^{16} + 1 6^{12} \cdot 2912 a_{0}^{16} - 1 6^{14} \cdot 13 a_{0}^{14}

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