Find dy/dx. x = t/6 + t , y = 6 + t

Solution:

x and y are the parametric functions with the parameter t.

Hence my parametric differentiation, we have dy/dx = dy/dt . dt/dx

dy/dt =  d(6 + t)/dt = 1

dx/dt = d(t/6 + t)/dt = 1/6 + 1 = 7/6

dy/dx = (dy/dt) / dx/dt  =(1/7)/6 = 6/7

dy/dx = 6/7

Let us take another example where 

x = t² + 2t  and y = t³ - 3t

dx/dt = 2t + 2 = 2(t + 1)

dy/dt = 3t² - 3 = 3(t² - 1)

dy/dx = (dy/dt) / dx/dt =   (3(t² - 1)) / 2(t + 1) = (3(t + 1)(t - 1)) / 2(t+1)

=  (3/2)(t - 1)

Find dy/dx. x = t/6 + t , y = 6 + t

Summary:

dy/dx = 6/7 when  x = t/6 + t , y = 6 + t