What is the tangent of (64x)/(x^2+64) ?

The tangent of (64x)/(x^2+64) is y=(64(-a_{0}^2+64))/((a_{0)^2+64)^2}x+(128a_{0}^3)/((a_{0)^2+64)^2}

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tangent of (64x)/(x^2+64)

tangent of

\frac{64 x}{x ^{2} + 64}

y = \frac{64 ( - a _{0}^{2} + 64 )}{( a _{0}^{2} + 64 ) ^{2}} x + \frac{128 a _{0}^{3}}{( a _{0}^{2} + 64 ) ^{2}}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, \frac{64 a _{0}}{a _{0}^{2} + 64})

Find the slope of

y = \frac{64 x}{x ^{2} + 64} : \frac{d y}{d x} = \frac{64 ( - x ^{2} + 64 )}{( x ^{2} + 64 ) ^{2}}

Compute the slope of

y = \frac{64 x}{x ^{2} + 64}

x = a_{0} : m = \frac{64 ( - a _{0}^{2} + 64 )}{( a _{0}^{2} + 64 ) ^{2}}

Find the line with slope m=

\frac{64 ( - a _{0}^{2} + 64 )}{( a _{0}^{2} + 64 ) ^{2}}

and passing through

(a_{0}, \frac{64 a _{0}}{a _{0}^{2} + 64}) : y = \frac{64 ( - a _{0}^{2} + 64 )}{( a _{0}^{2} + 64 ) ^{2}} x + \frac{128 a _{0}^{3}}{( a _{0}^{2} + 64 ) ^{2}}

y = \frac{64 ( - a _{0}^{2} + 64 )}{( a _{0}^{2} + 64 ) ^{2}} x + \frac{128 a _{0}^{3}}{( a _{0}^{2} + 64 ) ^{2}}

\frac{\partial}{\partial y} (2 x^{3} - x^{2} y)

\int (t^{2} - 1) (4 + 3 t) d t

f (x) = x^{4}, at x = 1

y = \frac{6}{x}, (25, \frac{6}{5})

\frac{x ^{2}}{7 + x}

What is the tangent of (64x)/(x^2+64) ?
The tangent of (64x)/(x^2+64) is y=(64(-a_{0}^2+64))/((a_{0)^2+64)^2}x+(128a_{0}^3)/((a_{0)^2+64)^2}