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

##### 1 Answer

Mar 22, 2017

# dy/dx = (2t^2(1 - 3e^(3t))e^(-t)) /((t-1) #

#### Explanation:

I assume that you want more than just the derivatives of the parametric equation, and actually require

We have;

# x(t) = e^t/(t+t) = e^t/(2t)#

Differentiating wrt

# dx/dt = {(2t)(e^t) - (2)(e^t)}/(2t)^2 #

# " " = (2(t-1)e^t)/(4t^2) #

# " " = ((t-1)e^t)/(2t^2) #

And:

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

Differentiating wrt

# dy/dt = 1 - 3e^(3t) #

And finally by the chain rule:

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

# " " = (1 - 3e^(3t)) / (((t-1)e^t)/(2t^2)) #

# " " = (1 - 3e^(3t)) * (2t^2)/((t-1)e^t) #

# " " = (2t^2(1 - 3e^(3t))e^(-t)) /((t-1) #