𝑦 = (𝑢 2 + 4𝑢 + 18) and 𝑢 = 𝑥 2 + 4 , find 𝑑𝑦

To find dy/dx, we first need to find dy/du and du/dx, then we can use the chain rule.

Given y = u^2 + 4u + 18, we can differentiate y with respect to u to get dy/du.

dy/du = 2u + 4

Given u = x^2 + 4, we can differentiate u with respect to x to get du/dx.

du/dx = 2x

Now we can use the chain rule, which states that dy/dx = dy/du * du/dx.

So, dy/dx = (2u + 4) * 2x

Substitute u = x^2 + 4 back into the equation to get dy/dx in terms of x.

dy/dx = (2(x^2 + 4) + 4) * 2x

Simplify to get the final answer:

dy/dx = 4x^3 + 16x + 8x = 4x^3 + 24x