Final answer:
The derivative (dy)/(dx) of the functions x=6t-5 and y=(2t-1)³ is found to be 2(2t - 1)² by using the chain rule and implicit differentiation.
Step-by-step explanation:
To find the derivative (dy)/(dx) for the functions x=6t-5 and y=(2t-1)³, we'll use the chain rule and the concept of implicit differentiation. Since both x and y are functions of t, we can find dx/dt and dy/dt first, and then divide dy/dt by dx/dt to get (dy)/(dx).
First, differentiate x with respect to t:
- dx/dt = d/dt (6t - 5) = 6
Then, differentiate y with respect to t using the chain rule:
- dy/dt = d/dt [(2t - 1)³] = 3(2t - 1)²(2) = 12(2t - 1)²
Now, we have dx/dt = 6 and dy/dt = 12(2t - 1)².
To find (dy)/(dx), we'll divide dy/dt by dx/dt:
(dy)/(dx) = (dy/dt) / (dx/dt) = [12(2t - 1)²] / 6 = 2(2t - 1)²