How do you differentiate the following parametric equation: # x(t)=tlnt, y(t)= t^3cost-tsin^2t #?

1 Answer
Feb 28, 2017

# dy/dx = (-t^3sint + 3t^2cost - tsin2t -sin^2t) /(1 + lnt) #

Explanation:

We have two parametric equations:

# x = tlnt #
# y = t^3cost - tsin^2t #

We differentiate wrt #t# (and apply the product rule):

# dx/dt = (t)(1/t) + (1)(lnt) #
# \ \ \ \ \ \ = 1 + lnt #

And:

# dy/dt = t^3cost - tsin^2t #
# \ \ \ \ \ \ = (t^3)(-sint) + (3t^2)(cost) - {(t)(2sintcost) + (1)(sin^2t)} #
# \ \ \ \ \ \ = -t^3sint + 3t^2cost - tsin2t -sin^2t #

And from the chain rule we have:

# dy/dx = (dy/dt)/(dy/dx) #
# \ \ \ \ \ \ = (-t^3sint + 3t^2cost - tsin2t -sin^2t) /(1 + lnt) #