If 𝑦 = 2𝑢 3 − 8𝑢 and 𝑢 = 7𝑥 − 𝑥 3 find 𝑑𝑦 𝑑�

The question seems to be asking for the derivative of y with respect to x, given the functions y = 2u^3 - 8u and u = 7x - x^3.

First, we need to find the derivative of y with respect to u (dy/du) and the derivative of u with respect to x (du/dx).

1. Find dy/du: The derivative of y = 2u^3 - 8u with respect to u is dy/du = 6u^2 - 8.

2. Find du/dx: The derivative of u = 7x - x^3 with respect to x is du/dx = 7 - 3x^2.

Then, we use the chain rule to find dy/dx. The chain rule states that dy/dx = dy/du * du/dx.

3. Find dy/dx: Multiply the derivatives we found in steps 1 and 2: dy/dx = (6u^2 - 8) * (7 - 3x^2).

However, this answer is in terms of both x and u. We want our final answer in terms of x only, so we need to substitute u = 7x - x^3 into our equation from step 3.

4. Substitute u into dy/dx: dy/dx = (6(7x - x^3)^2 - 8) * (7 - 3x^2).

This is the derivative of y with respect to x.