What is the tangent of ((3x))/((1+x^2)) ?

The tangent of ((3x))/((1+x^2)) is y=(3(1-a_{0}^2))/((1+a_{0)^2)^2}x+(6a_{0}^3)/((a_{0)^2+1)^2}

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tangent of ((3x))/((1+x^2))

tangent of

\frac{( 3 x )}{( 1 + x ^{2} )}

y = \frac{3 ( 1 - a _{0}^{2} )}{( 1 + a _{0}^{2} ) ^{2}} x + \frac{6 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{3 a _{0}}{1 + a _{0}^{2}})

Find the slope of

y = \frac{3 x}{1 + x ^{2}} : \frac{d y}{d x} = \frac{3 ( 1 - x ^{2} )}{( 1 + x ^{2} ) ^{2}}

Compute the slope of

y = \frac{3 x}{1 + x ^{2}}

x = a_{0} : m = \frac{3 ( 1 - a _{0}^{2} )}{( 1 + a _{0}^{2} ) ^{2}}

Find the line with slope m=

\frac{3 ( 1 - a _{0}^{2} )}{( 1 + a _{0}^{2} ) ^{2}}

and passing through

(a_{0}, \frac{3 a _{0}}{1 + a _{0}^{2}}) : y = \frac{3 ( 1 - a _{0}^{2} )}{( 1 + a _{0}^{2} ) ^{2}} x + \frac{6 a _{0}^{3}}{( a _{0}^{2} + 1 ) ^{2}}

y = \frac{3 ( 1 - a _{0}^{2} )}{( 1 + a _{0}^{2} ) ^{2}} x + \frac{6 a _{0}^{3}}{( a _{0}^{2} + 1 ) ^{2}}

\int tan (5 v) sec (5 v) d v

\frac{d y}{d t} = 20 - \frac{y}{230 + t}

\frac{1}{( x + 3 ) ^{2}}

\int tanh (x) d x

\frac{\partial}{\partial z} (x y z^{2})

What is the tangent of ((3x))/((1+x^2)) ?
The tangent of ((3x))/((1+x^2)) is y=(3(1-a_{0}^2))/((1+a_{0)^2)^2}x+(6a_{0}^3)/((a_{0)^2+1)^2}