Find dy/dx. x = t/8 + t , y = 8 + t

Solution:

x = t/8 + t

y = 8 + t

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

Hence we differentiate the functions as dy/dx = dy/dt . dt/dx

dx/dt  = d( t/8 + t) / dt = 1/8 + 1 = 9/8

dy/dt  =  d(8 + t) / dt =  1

dy/dx  =  (dy/dt) /  dx/dt= 1/(9/8)

=  8/9

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)

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Find dy/dx. x = t/8 + t , y = 8 + t

Summary:

dy/dx = 8/9 if  x = t/8 + t , y = 8 + t.