find the differential coefficient of y = (4t^3 - 3t)^6

To find the differential coefficient of the function y = (4t^3 - 3t)^6, we will use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 1: Identify the outer and inner functions. Here, the outer function is y = u^6 and the inner function is u = 4t^3 - 3t.

Step 2: Find the derivative of the outer function. The derivative of y = u^6 with respect to u is dy/du = 6u^5.

Step 3: Find the derivative of the inner function. The derivative of u = 4t^3 - 3t with respect to t is du/dt = 12t^2 - 3.

Step 4: Apply the chain rule. The chain rule states that dy/dt = dy/du * du/dt. Substituting the derivatives we found in steps 2 and 3, we get dy/dt = 6u^5 * (12t^2 - 3).

Step 5: Substitute u back into the equation. Remember that u = 4t^3 - 3t. Substituting this back into the equation gives us dy/dt = 6(4t^3 - 3t)^5 * (12t^2 - 3).

So, the differential coefficient of y = (4t^3 - 3t)^6 is dy/dt = 6(4t^3 - 3t)^5 * (12t^2 - 3).