Derivatives by different methods
a. Calculate d/dx (x²+x)² using the Chain Rule. Simplify your answer.

Verified step by step guidance

1. Step 1: Identify the outer and inner functions. Here, the outer function is \( u^2 \) and the inner function is \( u = x^2 + x \).
2. Step 2: Apply the Chain Rule, which states that \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \). In this case, \( f(u) = u^2 \) and \( g(x) = x^2 + x \).
3. Step 3: Differentiate the outer function \( f(u) = u^2 \) with respect to \( u \), which gives \( f'(u) = 2u \).
4. Step 4: Differentiate the inner function \( g(x) = x^2 + x \) with respect to \( x \), which gives \( g'(x) = 2x + 1 \).
5. Step 5: Combine the derivatives using the Chain Rule: \( \frac{d}{dx}[(x^2 + x)^2] = 2(x^2 + x) \cdot (2x + 1) \). Simplify the expression by distributing and combining like terms.