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Find dx/dt and dy/dx = 3t² + 5t,
X= 3t²
y = 6t - 4t²

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Final answer:

To find dx/dt and dy/dx given the equations X=3t² and y=6t-4t², we differentiate X with respect to t to get dx/dt=6t, and differentiate y to get dy/dt=6-8t. Then, using the chain rule, we find dy/dx by dividing dy/dt by dx/dt, resulting in dy/dx=(6-8t)/(6t).

Step-by-step explanation:

To find dx/dt and dy/dx given the equations X = 3t² and y = 6t - 4t² and the derivative dy/dx = 3t² + 5t, we will need to use calculus methods to differentiate the equations with respect to t and then apply the chain rule to find dy/dx.

Step 1: Differentiate X with respect to t

The equation X = 3t² is differentiated with respect to t to get dx/dt. X's derivative with respect to t is dx/dt = 6t, which represents the velocity of the function X(t) in terms of t.

Step 2: Differentiate y with respect to t

The equation y = 6t - 4t² is differentiated with respect to t to get dy/dt. The derivative is dy/dt = 6 - 8t.

Step 3: Find dy/dx using the chain rule

Using the chain rule, dy/dx = (dy/dt) / (dx/dt). Substituting the derivatives from the first two steps, we get dy/dx = (6 - 8t) / (6t).

Note that there's a discrepancy in the information provided - while the question states dy/dx = 3t² + 5t, the derived result from the given equations doesn't match this expression. Please double-check the given functions and the context of the problem to ensure the correct derivatives are found.

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