1004.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)

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

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

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

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

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

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

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

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

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

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