Let’s simplify how to understand the chain rule for derivatives with an example.

1. Find the Outer and Inner Functions: For the function (f(x) = (3x + 5)^2), we can break it down like this:

   • Inner function: (u = 3x + 5)
   • Outer function: (f(u) = u^2)

2. Calculate the Derivatives:

   • The derivative of the outer function is (f'(u) = 2u).
   • The derivative of the inner function is (u' = 3).

3. Use the Chain Rule: The chain rule tells us that to find the derivative (f'(x)), we use this formula: [f'(x) = f'(u) \cdot u'] Now, let’s plug in the derivatives: [f'(x) = 2u \cdot 3 = 2(3x + 5) \cdot 3.]

4. Simplify the Expression: If we simplify it, we get: [f'(x) = 6(3x + 5).]

And that’s all there is to it! Just remember: you multiply the outer function’s derivative by the inner function’s derivative!