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

##### 1 Answer

Aug 6, 2017

# dy/dx = (cost -tsint - 2tsintcost-sin^2t) / (e^t(sint+cost)) #

#### Explanation:

We have:

# x = e^tsint #

# y = tcost-tsin^2t #

Differentiating wrt

# dx/dt = (e^t)(d/tsint) + (d/dte^t)(sint) #

# " " = (e^t)(cost) + (e^t)(sint) #

# " " = e^t(sint+cost) #

# dy/dt = (t)(d/dtcost) + (d/dtt)(cost) - {(t)(d/dtsin^2t)+(d/dtt)(sin^2t) #

# " " = (t)(-sint) + (1)(cost) - {(t)(2sintcost)+(1)(sin^2t) #

# " " = -tsint + cost - 2tsintcost-sin^2t #

By the chain rule we have:

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

# " " = (cost -tsint - 2tsintcost-sin^2t) / (e^t(sint+cost)) #