What is the tangent of x^4-18x^2+81 ?

The tangent of x^4-18x^2+81 is y=(4a_{0}^3-36a_{0})x-3a_{0}^4+18a_{0}^2+81

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tangent of x^4-18x^2+81

tangent of

x^{4} - 18 x^{2} + 81

y = (4 a_{0}^{3} - 36 a_{0}) x - 3 a_{0}^{4} + 18 a_{0}^{2} + 81

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, a_{0}^{4} - 18 a_{0}^{2} + 81)

Find the slope of

y = x^{4} - 18 x^{2} + 81 : \frac{d y}{d x} = 4 x^{3} - 36 x

Compute the slope of

y = x^{4} - 18 x^{2} + 81

x = a_{0} : m = 4 a_{0}^{3} - 36 a_{0}

Find the line with slope m=

4 a_{0}^{3} - 36 a_{0}

and passing through

(a_{0}, a_{0}^{4} - 18 a_{0}^{2} + 81) : y = (4 a_{0}^{3} - 36 a_{0}) x - 3 a_{0}^{4} + 18 a_{0}^{2} + 81

y = (4 a_{0}^{3} - 36 a_{0}) x - 3 a_{0}^{4} + 18 a_{0}^{2} + 81

\int \frac{x ^{2}}{y} d x

\frac{d y}{d x} = (3 e x - 9 e - x)^{2}

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

\frac{d y}{d x} = 3 x^{- 2} e^{- y}

\int \frac{( x ^{2} + 9 x + 1 )}{( x ^{2} + 1 ) ^{2}} d x

What is the tangent of x^4-18x^2+81 ?
The tangent of x^4-18x^2+81 is y=(4a_{0}^3-36a_{0})x-3a_{0}^4+18a_{0}^2+81