How do you differentiate the following parametric equation: # x(t)=tlnt, y(t)= cost-tsin^2t #?

1 Answer
Jul 13, 2016

# (df(t))/dt = (ln(t) + 1 , -sin(t) - sin^2(t) - 2tsin(t)cos(t))#

Explanation:

Differentiating a parametric equation is as easy as differentiating each individual equation for its components.

If #f(t) = (x(t), y(t))# then #(df(t))/dt = ((dx(t))/dt , (dy(t))/dt)#

So we first determine our component derivatives:
# (dx(t))/dt = ln(t) + t/t = ln(t) + 1#
# (dy(t))/dt = -sin(t) - sin^2(t) - 2tsin(t)cos(t)#

Therefore the final parametric curve's derivatives is simply a vector of the derivatives:
#(df(t))/dt = ((dx(t))/dt , (dy(t))/dt)#
# = (ln(t) + 1 , -sin(t) - sin^2(t) - 2tsin(t)cos(t))#