What is the tangent of y=(x^3-25x)^{14} ?

The tangent of y=(x^3-25x)^{14} is

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tangent of y=(x^3-25x)^{14}

tangent of

y = (x^{3} - 25 x)^{14}

y = 14 (a_{0}^{3} - 25 a_{0})^{13} (3 a_{0}^{2} - 25) x - 41 a_{0}^{42} + 13650 a_{0}^{40} - 2104375 a_{0}^{38} + 199062500 a_{0}^{36} - 12903515625 a_{0}^{34} + 606074218750 a_{0}^{32} - 2.12615 E 13 a_{0}^{30} + 2 5^{7} \cdot 68640 a_{0}^{28} + 1.46631 E 14 a_{0}^{28} - 2 5^{8} \cdot 51051 a_{0}^{26} - 2 5^{7} \cdot 600600 a_{0}^{26} + 2 5^{9} \cdot 28028 a_{0}^{24} + 2 5^{8} \cdot 450450 a_{0}^{24} - 2 5^{10} \cdot 11011 a_{0}^{22} - 2 5^{9} \cdot 250250 a_{0}^{22} + 2 5^{11} \cdot 2912 a_{0}^{20} + 2 5^{10} \cdot 100100 a_{0}^{20} - 2 5^{12} \cdot 455 a_{0}^{18} - 2 5^{11} \cdot 27300 a_{0}^{18} + 2 5^{13} \cdot 28 a_{0}^{16} + 2 5^{12} \cdot 4550 a_{0}^{16} - 2 5^{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} - 25 a_{0})^{14})

Find the slope of

y = (x^{3} - 25 x)^{14} : \frac{d y}{d x} = 14 (x^{3} - 25 x)^{13} (3 x^{2} - 25)

Compute the slope of

y = (x^{3} - 25 x)^{14}

x = a_{0} : m = 14 (a_{0}^{3} - 25 a_{0})^{13} (3 a_{0}^{2} - 25)

Find the line with slope m=

14 (a_{0}^{3} - 25 a_{0})^{13} (3 a_{0}^{2} - 25)

and passing through

(a_{0}, (a_{0}^{3} - 25 a_{0})^{14}) : y = 14 (a_{0}^{3} - 25 a_{0})^{13} (3 a_{0}^{2} - 25) x - 41 a_{0}^{42} + 13650 a_{0}^{40} - 2104375 a_{0}^{38} + 199062500 a_{0}^{36} - 12903515625 a_{0}^{34} + 606074218750 a_{0}^{32} - 2.12615 E 13 a_{0}^{30} + 2 5^{7} \cdot 68640 a_{0}^{28} + 1.46631 E 14 a_{0}^{28} - 2 5^{8} \cdot 51051 a_{0}^{26} - 2 5^{7} \cdot 600600 a_{0}^{26} + 2 5^{9} \cdot 28028 a_{0}^{24} + 2 5^{8} \cdot 450450 a_{0}^{24} - 2 5^{10} \cdot 11011 a_{0}^{22} - 2 5^{9} \cdot 250250 a_{0}^{22} + 2 5^{11} \cdot 2912 a_{0}^{20} + 2 5^{10} \cdot 100100 a_{0}^{20} - 2 5^{12} \cdot 455 a_{0}^{18} - 2 5^{11} \cdot 27300 a_{0}^{18} + 2 5^{13} \cdot 28 a_{0}^{16} + 2 5^{12} \cdot 4550 a_{0}^{16} - 2 5^{14} \cdot 13 a_{0}^{14}

y = 14 (a_{0}^{3} - 25 a_{0})^{13} (3 a_{0}^{2} - 25) x - 41 a_{0}^{42} + 13650 a_{0}^{40} - 2104375 a_{0}^{38} + 199062500 a_{0}^{36} - 12903515625 a_{0}^{34} + 606074218750 a_{0}^{32} - 2.12615 E 13 a_{0}^{30} + 2 5^{7} \cdot 68640 a_{0}^{28} + 1.46631 E 14 a_{0}^{28} - 2 5^{8} \cdot 51051 a_{0}^{26} - 2 5^{7} \cdot 600600 a_{0}^{26} + 2 5^{9} \cdot 28028 a_{0}^{24} + 2 5^{8} \cdot 450450 a_{0}^{24} - 2 5^{10} \cdot 11011 a_{0}^{22} - 2 5^{9} \cdot 250250 a_{0}^{22} + 2 5^{11} \cdot 2912 a_{0}^{20} + 2 5^{10} \cdot 100100 a_{0}^{20} - 2 5^{12} \cdot 455 a_{0}^{18} - 2 5^{11} \cdot 27300 a_{0}^{18} + 2 5^{13} \cdot 28 a_{0}^{16} + 2 5^{12} \cdot 4550 a_{0}^{16} - 2 5^{14} \cdot 13 a_{0}^{14}

f (x) = (8 - x)^{6}

x \to 0 + lim (\frac{- 1}{x})

\frac{16 s + 5}{s ^{2} + 9}

(f (x))^{2}

y = (2 - \frac{7}{5}) e^{3 t} + \frac{7}{5} e^{- 2 t}