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

##### 1 Answer

Apr 21, 2018

First, differentiate each individual function as you have been all year.

using the chain rule,

#x(t)=(t-2)^(1/2)#

#" "" "=>dx/dt=1/2(t-2)^(-1/2)*d/dt(t-2)#

#" "" "color(white)(=>dx/dt)=1/2(t-2)^(-1/2)*1#

#" "" "color(white)(=>dx/dt)=color(red)(1/(2sqrt(t-2))#

and, by the product rule,

#y(t)=t^2-2t^3e^t#

#" "" "=>dy/dt=2t-[(d/dt2t^3)e^t-2t^3(d/dte^t)]#

#" "" "color(white)(=>dy/dt)=2t-[6t^2e^t-2t^3e^t]#

#" "" "color(white)(=>dy/dt)=color(red)(2t(1-3te^t+t^2e^t)#

if we want to know

#dy/dx=(dy/dt)/(dx/dt)=(2t(1-3te^t+t^2e^t))/(1/(2sqrt(t-2)))=color(blue)(4tsqrt(t-2)(1-3te^t+t^2e^t)#