Use the chain rule to solve the following: a. 𝑦 = (𝑢 2 + 4𝑢 + 18) and 𝑢 = 𝑥 2 + 4 , find 𝑑𝑦 𝑑�

The chain rule in calculus is a method used to compute the derivative of a composite function. 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.

Given the functions y = u^2 + 4u + 18 and u = x^2 + 4, we want to find dy/dx.

Step 1: Find dy/du and du/dx

The derivative of y with respect to u, dy/du, is 2u + 4.

The derivative of u with respect to x, du/dx, is 2x.

Step 2: Apply the chain rule

The chain rule states that dy/dx = dy/du * du/dx.

Substituting the derivatives we found in step 1, we get dy/dx = (2u + 4) * 2x.

Step 3: Substitute u back into the equation

We know that u = x^2 + 4, so we substitute this back into the equation to get dy/dx in terms of x.

dy/dx = (2(x^2 + 4) + 4) * 2x = 4x^3 + 16x + 8x = 4x^3 + 24x.

So, the derivative of y with respect to x, dy/dx, is 4x^3 + 24x.