What is the tangent of y=(x^3-16x)^{12} ?

The tangent of y=(x^3-16x)^{12} is

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tangent of y=(x^3-16x)^{12}

tangent of

y = (x^{3} - 16 x)^{12}

y = 12 (a_{0}^{3} - 16 a_{0})^{11} (3 a_{0}^{2} - 16) x - 35 a_{0}^{36} + 6336 a_{0}^{34} - 523776 a_{0}^{32} + 26132480 a_{0}^{30} - 875888640 a_{0}^{28} + 20761804800 a_{0}^{26} - 356549394432 a_{0}^{24} + 4464618504192 a_{0}^{22} - 1 6^{8} \cdot 5445 a_{0}^{20} - 1.70081 E 13 a_{0}^{20} + 1 6^{9} \cdot 1760 a_{0}^{18} + 1 6^{8} \cdot 31680 a_{0}^{18} - 1 6^{10} \cdot 330 a_{0}^{16} - 1 6^{9} \cdot 10560 a_{0}^{16} + 1 6^{11} \cdot 24 a_{0}^{14} + 1 6^{10} \cdot 2112 a_{0}^{14} - 1 6^{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} - 16 a_{0})^{12})

Find the slope of

y = (x^{3} - 16 x)^{12} : \frac{d y}{d x} = 12 (x^{3} - 16 x)^{11} (3 x^{2} - 16)

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} - 16 a_{0})^{11} (3 a_{0}^{2} - 16)

Find the line with slope m=

12 (a_{0}^{3} - 16 a_{0})^{11} (3 a_{0}^{2} - 16)

and passing through

(a_{0}, (a_{0}^{3} - 16 a_{0})^{12}) : y = 12 (a_{0}^{3} - 16 a_{0})^{11} (3 a_{0}^{2} - 16) x - 35 a_{0}^{36} + 6336 a_{0}^{34} - 523776 a_{0}^{32} + 26132480 a_{0}^{30} - 875888640 a_{0}^{28} + 20761804800 a_{0}^{26} - 356549394432 a_{0}^{24} + 4464618504192 a_{0}^{22} - 1 6^{8} \cdot 5445 a_{0}^{20} - 1.70081 E 13 a_{0}^{20} + 1 6^{9} \cdot 1760 a_{0}^{18} + 1 6^{8} \cdot 31680 a_{0}^{18} - 1 6^{10} \cdot 330 a_{0}^{16} - 1 6^{9} \cdot 10560 a_{0}^{16} + 1 6^{11} \cdot 24 a_{0}^{14} + 1 6^{10} \cdot 2112 a_{0}^{14} - 1 6^{12} \cdot 11 a_{0}^{12}

y = 12 (a_{0}^{3} - 16 a_{0})^{11} (3 a_{0}^{2} - 16) x - 35 a_{0}^{36} + 6336 a_{0}^{34} - 523776 a_{0}^{32} + 26132480 a_{0}^{30} - 875888640 a_{0}^{28} + 20761804800 a_{0}^{26} - 356549394432 a_{0}^{24} + 4464618504192 a_{0}^{22} - 1 6^{8} \cdot 5445 a_{0}^{20} - 1.70081 E 13 a_{0}^{20} + 1 6^{9} \cdot 1760 a_{0}^{18} + 1 6^{8} \cdot 31680 a_{0}^{18} - 1 6^{10} \cdot 330 a_{0}^{16} - 1 6^{9} \cdot 10560 a_{0}^{16} + 1 6^{11} \cdot 24 a_{0}^{14} + 1 6^{10} \cdot 2112 a_{0}^{14} - 1 6^{12} \cdot 11 a_{0}^{12}

5 x - 2 x^{2}

x \to - 5 - lim (\frac{x + 5}{x + 5})

\int_{6}^{7} \frac{1}{x ^{2} - 1} d x

f (x) = 3 x - 7

\int \frac{1}{36 e ^{- 7 x} + e ^{7 x}} d x