Final answer:
To find dy/dx for the given functions, we apply the chain rule which results in dy/dx equals -36x², corresponding with option b.Option B is the correct answer.
Step-by-step explanation:
To find dy/dx for the given functions y = 4u³ and u = -3x, we use Leibniz's notation for the chain rule, which states that to find the derivative of a composite function, we take the derivative of the outer function with respect to the inner function (dy/du) and multiply it by the derivative of the inner function with respect to x (du/dx).
First, we find the derivative dy/du for
y = 4u³:
dy/du = 12u²
Next, we calculate du/dx for u = -3x:
du/dx = -3
Now using the chain rule: dy/dx = dy/du × du/dx
Substituting the derivatives we found:
dy/dx = (12u²) × (-3)
As we know u in terms of x (u = -3x), we substitute u to express dy/dx entirely in terms of x:
dy/dx = 12(-3x)² × (-3) = 12(9x²) × (-3) = -108x²
Finally, after simplifying, we get the correct answer:
dy/dx = -36x²