\( 6 \quad \) b) \( y=\sqrt{\left(x^{3}+2 x\right)} \) \( \frac{d u^{n}}{d x}=n=\left(x^{3}+2 x\right)^{\frac{1}{2}} \frac{d y}{d x}= \) \( \frac{d u}{d x} \)

To find the derivative \( \frac{dy}{dx} \) for the function \( y = \sqrt{x^3 + 2x} \), you would apply the chain rule. First, rewrite the function as \( y = (x^3 + 2x)^{1/2} \). Then, using the chain rule: \[ \frac{dy}{dx} = \frac{1}{2}(x^3 + 2x)^{-1/2} \cdot \frac{d}{dx}(x^3 + 2x). \] Next, calculate \( \frac{d}{dx}(x^3 + 2x) = 3x^2 + 2 \). Plugging this in gives: \[ \frac{dy}{dx} = \frac{1}{2}(x^3 + 2x)^{-1/2} \cdot (3x^2 + 2). \] Now simplify it to get the final expression for the derivative! While the math may seem daunting, it's all about staying organized. Breaking it down step by step can prevent mistakes. A common pitfall is forgetting to apply the chain rule properly. Be sure to differentiate the inner function (inside the square root) correctly and don't skip simplifying. For those wanting to dive deeper into calculus, consider exploring resources like "Calculus Made Easy" by Silvanus P. Thompson, which breaks down complex concepts into easier-to-understand pieces. Websites like Khan Academy also offer engaging video tutorials to solidify your understanding and provide interactive exercises for practice.