What is the tangent of y=(x^3-9x)^{12} ?

The tangent of y=(x^3-9x)^{12} is

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tangent of y=(x^3-9x)^{12}

tangent of

y = (x^{3} - 9 x)^{12}

y = 12 (a_{0}^{3} - 9 a_{0})^{11} (3 a_{0}^{2} - 9) x - 35 a_{0}^{36} + 3564 a_{0}^{34} - 165726 a_{0}^{32} + 4651020 a_{0}^{30} - 87687765 a_{0}^{28} + 1169170200 a_{0}^{26} - 11294184132 a_{0}^{24} + 79550340408 a_{0}^{22} - 404854411005 a_{0}^{20} + 1448952628860 a_{0}^{18} - 9^{10} \cdot 330 a_{0}^{16} - 2301277704660 a_{0}^{16} + 9^{11} \cdot 24 a_{0}^{14} + 9^{10} \cdot 1188 a_{0}^{14} - 9^{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} - 9 a_{0})^{12})

Find the slope of

y = (x^{3} - 9 x)^{12} : \frac{d y}{d x} = 12 (x^{3} - 9 x)^{11} (3 x^{2} - 9)

Compute the slope of

y = (x^{3} - 9 x)^{12}

x = a_{0} : m = 12 (a_{0}^{3} - 9 a_{0})^{11} (3 a_{0}^{2} - 9)

Find the line with slope m=

12 (a_{0}^{3} - 9 a_{0})^{11} (3 a_{0}^{2} - 9)

and passing through

(a_{0}, (a_{0}^{3} - 9 a_{0})^{12}) : y = 12 (a_{0}^{3} - 9 a_{0})^{11} (3 a_{0}^{2} - 9) x - 35 a_{0}^{36} + 3564 a_{0}^{34} - 165726 a_{0}^{32} + 4651020 a_{0}^{30} - 87687765 a_{0}^{28} + 1169170200 a_{0}^{26} - 11294184132 a_{0}^{24} + 79550340408 a_{0}^{22} - 404854411005 a_{0}^{20} + 1448952628860 a_{0}^{18} - 9^{10} \cdot 330 a_{0}^{16} - 2301277704660 a_{0}^{16} + 9^{11} \cdot 24 a_{0}^{14} + 9^{10} \cdot 1188 a_{0}^{14} - 9^{12} \cdot 11 a_{0}^{12}

y = 12 (a_{0}^{3} - 9 a_{0})^{11} (3 a_{0}^{2} - 9) x - 35 a_{0}^{36} + 3564 a_{0}^{34} - 165726 a_{0}^{32} + 4651020 a_{0}^{30} - 87687765 a_{0}^{28} + 1169170200 a_{0}^{26} - 11294184132 a_{0}^{24} + 79550340408 a_{0}^{22} - 404854411005 a_{0}^{20} + 1448952628860 a_{0}^{18} - 9^{10} \cdot 330 a_{0}^{16} - 2301277704660 a_{0}^{16} + 9^{11} \cdot 24 a_{0}^{14} + 9^{10} \cdot 1188 a_{0}^{14} - 9^{12} \cdot 11 a_{0}^{12}

x \to \frac{x}{8} lim (tan (24 \cdot x))

\int csc^{4} (4 x) d x

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\int \frac{100}{25 + ( 5 y ) ^{2}} + e^{4} d y

\frac{y ( 4 x + 4 y )}{( x ^{2} + y ^{2} )}