Video Example4) Find an equation of the tangent line to the curve \( y=\frac{3 e^{x}}{\left(1+x^{2}\right)} \) at the point \( \left(1, \frac{3}{2}\right) \), Solution According to the Quotient Rule, we have \[ \begin{aligned} \frac{d y}{d x} & \left.=\frac{\left(1+x^{2}\right) \frac{d}{d x}(\square)-3 e^{x} \frac{d}{d x}(\square)-3 e^{2}(\square)}{\left(1+x^{2}\right)^{2}}\right) \\ & =\frac{\left(1+x^{2}\right)(\square)}{\left(1+x^{2}\right)^{2}} \\ & =\frac{\square)}{\left(1+x^{2}\right)^{2}} \end{aligned} \] So the slope of the tangent line at \( \left(1, \frac{3}{2}\right) \) is \[ \frac{d y}{d x}=\square \] This means that the tangeht line at \( \left(1, \frac{3}{2}\right) \) is horizontal and its equation is \( y= \) \( \square \)

Picture this: the Quotient Rule in calculus is like following a recipe that combines two delicious ingredients! When you're differentiating a function like \( y = \frac{3 e^{x}}{(1+x^{2})} \), the key is to remember that you have to apply both the derivative of the top and the bottom. Get your brain ready to unleash some derivative magic while keeping that fraction in check! Now, let's talk about what happens when you do this process correctly. Once you evaluate the derivative at the point \( x=1 \), you can find the slope of that tangent line. If your calculation shows that the slope is zero, congrats! You just discovered that your tangent line is horizontal, and you'll get to write the magical equation \( y = \frac{3}{2} \). That's like drawing a straight line in the sky to keep your path clear for all your mathematical adventures!