Chain Rule Mastery

For algebraic functions: $y = (g(x))^n$

1. Visualize the Concept

The Chain Rule says: Differentiate the Outside (keep inside the same), then multiply by the derivative of the Inside.

2. Decompose: Inner & Outer

To solve $y = (3x^2 + 5)^4$, drag the correct terms to their boxes.

3x^2 + 5
6x
u^4
4u^3
4(3x^2+5)
3x^2
1. Inner Function ($u$)
2. Outer Function ($y=u^n$)
3. Derivative of Inner ($u'$)
4. Derivative of Outer ($f'(u)$)

3. Step-by-Step Practice

Solve the derivative for the random function below.

$$ f(x) = (2x + 1)^3 $$
Step 1: Differentiate the Outside (Apply Power Rule, keep inside same).

$f'(outer) = $ $\times ( \text{Inside} )$
Step 2: Differentiate the Inside.

$\frac{d}{dx}(\text{Inside}) = $
Step 3: Final Answer (Multiply Step 1 coeff by Step 2).

$f'(x) = $ $( \text{Inside} )^{\text{power}}$