1. **Problem Statement:** Find the derivative of a composite function using the chain rule. 2. **Formula:** The chain rule states that if you have a composite function $y = f(g(x))$, then its derivative is given by: $$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$$ This means you first differentiate the outer function evaluated at the inner function, then multiply by the derivative of the inner function. 3. **Explanation:** The chain rule is used when one function is inside another. For example, if $y = (3x^2 + 2)^5$, the outer function is $f(u) = u^5$ and the inner function is $g(x) = 3x^2 + 2$. 4. **Step-by-step solution:** - Identify the outer function $f(u) = u^5$ and inner function $g(x) = 3x^2 + 2$. - Compute the derivative of the outer function: $f'(u) = 5u^4$. - Compute the derivative of the inner function: $g'(x) = 6x$. - Apply the chain rule: $$\frac{dy}{dx} = f'(g(x)) \cdot g'(x) = 5(3x^2 + 2)^4 \cdot 6x$$ 5. **Final answer:** $$\frac{dy}{dx} = 30x(3x^2 + 2)^4$$ This is the derivative of the composite function using the chain rule.