What is the tangent of y=(-4x)/(x^2+1) ?

The tangent of y=(-4x)/(x^2+1) is y=-(4(-a_{0}^2+1))/((a_{0)^2+1)^2}x-(8a_{0}^3)/((a_{0)^2+1)^2}

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tangent of y=(-4x)/(x^2+1)

tangent of

y = \frac{- 4 x}{x ^{2} + 1}

y = - \frac{4 ( - a _{0}^{2} + 1 )}{( a _{0}^{2} + 1 ) ^{2}} x - \frac{8 a _{0}^{3}}{( a _{0}^{2} + 1 ) ^{2}}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, - \frac{4 a _{0}}{a _{0}^{2} + 1})

Find the slope of

y = \frac{- 4 x}{x ^{2} + 1} : \frac{d y}{d x} = - \frac{4 ( - x ^{2} + 1 )}{( x ^{2} + 1 ) ^{2}}

Compute the slope of

y = \frac{- 4 x}{x ^{2} + 1}

x = a_{0} : m = - \frac{4 ( - a _{0}^{2} + 1 )}{( a _{0}^{2} + 1 ) ^{2}}

Find the line with slope m=

- \frac{4 ( - a _{0}^{2} + 1 )}{( a _{0}^{2} + 1 ) ^{2}}

and passing through

(a_{0}, - \frac{4 a _{0}}{a _{0}^{2} + 1}) : y = - \frac{4 ( - a _{0}^{2} + 1 )}{( a _{0}^{2} + 1 ) ^{2}} x - \frac{8 a _{0}^{3}}{( a _{0}^{2} + 1 ) ^{2}}

y = - \frac{4 ( - a _{0}^{2} + 1 )}{( a _{0}^{2} + 1 ) ^{2}} x - \frac{8 a _{0}^{3}}{( a _{0}^{2} + 1 ) ^{2}}

5 x^{3} - 6 x^{2} + 12 x - 3, 3

\frac{d}{d x} (4 x^{3} + 3 x^{2} - 6 x + 1)

x \to 0 + lim (\frac{4}{x ^{2} - x})

\frac{\partial}{\partial y} (y e^{x y} + y)

x^{3} - 5 x^{2} + 13

What is the tangent of y=(-4x)/(x^2+1) ?
The tangent of y=(-4x)/(x^2+1) is y=-(4(-a_{0}^2+1))/((a_{0)^2+1)^2}x-(8a_{0}^3)/((a_{0)^2+1)^2}