1005.sagews

t = var('t')
x = 80* cos(1080*t) * sin( atan(2*t))

dx = diff(x,t)
ddx = diff(dx,t)
show(dx)
show(ddx)

-172800 \, \frac{t \sin\left(1080 \, t\right)}{\sqrt{4 \, t^{2} + 1}} - 640 \, \frac{t^{2} \cos\left(1080 \, t\right)}{{(4 \, t^{2} + 1)}^{\left(\frac{3}{2}\right)}} + 160 \, \frac{\cos\left(1080 \, t\right)}{\sqrt{4 \, t^{2} + 1}}

-186624000 \, \frac{t \cos\left(1080 \, t\right)}{\sqrt{4 \, t^{2} + 1}} + 1382400 \, \frac{t^{2} \sin\left(1080 \, t\right)}{{(4 \, t^{2} + 1)}^{\left(\frac{3}{2}\right)}} + 7680 \, \frac{t^{3} \cos\left(1080 \, t\right)}{{(4 \, t^{2} + 1)}^{\left(\frac{5}{2}\right)}} - 345600 \, \frac{\sin\left(1080 \, t\right)}{\sqrt{4 \, t^{2} + 1}} - 1920 \, \frac{t \cos\left(1080 \, t\right)}{{(4 \, t^{2} + 1)}^{\left(\frac{3}{2}\right)}}

t = var('t')
y = 80* sin(1080*t) * sin( atan(2*t))

dy = diff(x,t)
ddy = diff(dx,t)
show(dy)
show(ddy)

-172800 \, \frac{t \sin\left(1080 \, t\right)}{\sqrt{4 \, t^{2} + 1}} - 640 \, \frac{t^{2} \cos\left(1080 \, t\right)}{{(4 \, t^{2} + 1)}^{\left(\frac{3}{2}\right)}} + 160 \, \frac{\cos\left(1080 \, t\right)}{\sqrt{4 \, t^{2} + 1}}

-186624000 \, \frac{t \cos\left(1080 \, t\right)}{\sqrt{4 \, t^{2} + 1}} + 1382400 \, \frac{t^{2} \sin\left(1080 \, t\right)}{{(4 \, t^{2} + 1)}^{\left(\frac{3}{2}\right)}} + 7680 \, \frac{t^{3} \cos\left(1080 \, t\right)}{{(4 \, t^{2} + 1)}^{\left(\frac{5}{2}\right)}} - 345600 \, \frac{\sin\left(1080 \, t\right)}{\sqrt{4 \, t^{2} + 1}} - 1920 \, \frac{t \cos\left(1080 \, t\right)}{{(4 \, t^{2} + 1)}^{\left(\frac{3}{2}\right)}}

t = var('t')
z = -5 - 120 * sin( atan(2*t) )

dz = diff(z,t)
ddz = diff(dz,t)
show(dz)
show(ddz)

960 \, \frac{t^{2}}{{(4 \, t^{2} + 1)}^{\left(\frac{3}{2}\right)}} - 240 \, \frac{1}{\sqrt{4 \, t^{2} + 1}}

-11520 \, \frac{t^{3}}{{(4 \, t^{2} + 1)}^{\left(\frac{5}{2}\right)}} + 2880 \, \frac{t}{{(4 \, t^{2} + 1)}^{\left(\frac{3}{2}\right)}}