What is the tangent of (3x)/(x^2-6) ?

The tangent of (3x)/(x^2-6) is y=(3(-a_{0}^2-6))/((a_{0)^2-6)^2}x+(6a_{0}^3)/(a_{0)^4-12a_{0}^2+36}

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tangent of (3x)/(x^2-6)

tangent of

\frac{3 x}{x ^{2} - 6}

y = \frac{3 ( - a _{0}^{2} - 6 )}{( a _{0}^{2} - 6 ) ^{2}} x + \frac{6 a _{0}^{3}}{a _{0}^{4} - 12 a _{0}^{2} + 36}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, \frac{3 a _{0}}{a _{0}^{2} - 6})

Find the slope of

y = \frac{3 x}{x ^{2} - 6} : \frac{d y}{d x} = \frac{3 ( - x ^{2} - 6 )}{( x ^{2} - 6 ) ^{2}}

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 = \frac{3 ( - a _{0}^{2} - 6 )}{( a _{0}^{2} - 6 ) ^{2}}

Find the line with slope m=

\frac{3 ( - a _{0}^{2} - 6 )}{( a _{0}^{2} - 6 ) ^{2}}

and passing through

(a_{0}, \frac{3 a _{0}}{a _{0}^{2} - 6}) : y = \frac{3 ( - a _{0}^{2} - 6 )}{( a _{0}^{2} - 6 ) ^{2}} x + \frac{6 a _{0}^{3}}{a _{0}^{4} - 12 a _{0}^{2} + 36}

y = \frac{3 ( - a _{0}^{2} - 6 )}{( a _{0}^{2} - 6 ) ^{2}} x + \frac{6 a _{0}^{3}}{a _{0}^{4} - 12 a _{0}^{2} + 36}

y = 5 + 5 x^{2} - 2 x^{3}

\frac{d}{d x} (e^{x} (1 + x))

f (x) = 5 x + 6

\frac{d}{d x} ((3 x^{3} + 4) (x^{4} - 4 x))

y = \frac{1}{y ^{2}}

What is the tangent of (3x)/(x^2-6) ?
The tangent of (3x)/(x^2-6) is y=(3(-a_{0}^2-6))/((a_{0)^2-6)^2}x+(6a_{0}^3)/(a_{0)^4-12a_{0}^2+36}