# How do you differentiate the following parametric equation: # x(t)=te^tsint, y(t)= tcost-sin^2t #?

##### 1 Answer

Mar 24, 2018

# dx/dt = te^tcost + te^tsint + e^tsint #

# dy/dt = -tsint + cost-2sintcost#

#### Explanation:

We have parametric equations:

# x(t) = te^t sint#

# y(t) = tcost -sin^2t#

So, using the product rule and differentiating wrt

# x'(t) = (t)(e^t)(d/dt sint) + (t)(d/dt e^t)(sint) + (d/dt t)(e^t)(sint) #

# \ \ \ \ \ \ \ = (te^t)(cost) + (t)(e^t)(sint) + (1)(e^tsint) #

# \ \ \ \ \ \ \ = te^tcost + te^tsint + e^tsint #

And:

# y'(t) = (t)(d/dt cost) + (d/dt t)(cost)-d/dt sin^2t#

# \ \ \ \ \ \ \ = (t)(-sint) + (1)(cost)-2sint(cost)#

# \ \ \ \ \ \ \ = -tsint + cost-2sintcost#

This technically is the solution, but more likely we seek the full derivative:

# dy/dx = (dy//dt)/(dx//dt) #

# \ \ \ \ \ \ = (cost-tsint -2sintcost)/(te^tcost + te^tsint + e^tsint) #