Differentlate the function. \[ y=\sqrt{3 x^{2}+2 x+7} \] \( \frac{d y}{d x}=6 x+2 \) Need Help? Road It Watch it

To differentiate the function \( y=\sqrt{3x^{2}+2x+7} \), we can use the chain rule. The outer function is \( \sqrt{u} \) where \( u = 3x^2 + 2x + 7 \). The derivative of \( \sqrt{u} \) is \( \frac{1}{2\sqrt{u}} \), and we also need to differentiate \( u \) with respect to \( x \). This gives: \[ \frac{du}{dx} = 6x + 2 \] Putting it all together, we have: \[ \frac{dy}{dx} = \frac{1}{2\sqrt{3x^2 + 2x + 7}} \cdot (6x + 2) \] So, the final answer is: \[ \frac{dy}{dx} = \frac{6x + 2}{2\sqrt{3x^2 + 2x + 7}} \] Understanding the chain rule can make differentiation feel like a breeze! It's all about breaking down the problem into smaller, manageable pieces. If you're gearing up for calculus, don't forget to brush up on the basics like limits and derivatives. They’re the bread and butter of understanding more complex concepts! Books like *Calculus by James Stewart* or *Calculus Made Easy by Silvanus P. Thompson* can provide a solid foundation and fun insights into the world of calculus.