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 $$ y= $$ $$ \square $$

Question

UpStudy AI · Accepted Answer

We are given the function
$$
y=\frac{3 e^x}{1+x^2}
$$
and we need to find the equation of the tangent line at the point $$\left(1, \frac{3}{2}\right)$$.

### Step 1. Compute the derivative

Let
$$
f(x)=3e^x \quad \text{and} \quad g(x)=1+x^2.
$$
Then, their derivatives are
$$
f'(x)=3e^x \quad \text{and} \quad g'(x)=2x.
$$

Using the Quotient Rule,
$$
\frac{dy}{dx}=\frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^2},
$$
we substitute $$f(x)$$, $$f'(x)$$, $$g(x)$$, and $$g'(x)$$:
$$
\frac{dy}{dx}=\frac{(1+x^2)(3e^x)-3e^x(2x)}{(1+x^2)^2}.
$$

### Step 2. Simplify the derivative

Factor $$3e^x$$ from the numerator:
$$
\frac{dy}{dx}=\frac{3e^x\left[(1+x^2)-2x\right]}{(1+x^2)^2}.
$$
The expression $$(1+x^2)-2x$$ simplifies to
$$
1+x^2-2x.
$$

Thus, the derivative is
$$
\frac{dy}{dx}=\frac{3e^x\left(1+x^2-2x\right)}{(1+x^2)^2}.
$$

### Step 3. Evaluate the derivative at $$x=1$$

Substitute $$x=1$$ into the derivative:
$$
\frac{dy}{dx}\Big|_{x=1}=\frac{3e^1\left(1+1^2-2\cdot 1\right)}{(1+1^2)^2}.
$$
Compute the expression inside the parentheses:
$$
1+1-2=0.
$$
So the numerator becomes
$$
3e \cdot 0 = 0.
$$
And the denominator is
$$
(1+1)^2 = 4.
$$
Thus,
$$
\frac{dy}{dx}\Big|_{x=1}=\frac{0}{4}=0.
$$

### Step 4. Write the equation of the tangent line

The slope $$m$$ of the tangent line at $$\left(1, \frac{3}{2}\right)$$ is $$0$$; hence, the tangent line is horizontal. Its equation is therefore
$$
y=\frac{3}{2}.
$$
The equation of the tangent line at the point $$\left(1, \frac{3}{2}\right)$$ is $$ y = \frac{3}{2} $$.

Video Example4)
Find an equation of the tangent line to the curve at the point ,
Solution
According to the Quotient Rule, we have

So the slope of the tangent line at is

This means that the tangeht line at is horizontal and its equation is

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Solución

Step 1. Compute the derivative

Step 2. Simplify the derivative

Step 3. Evaluate the derivative at

Step 4. Write the equation of the tangent line

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Video Example4) Find an equation of the tangent line to the curve at the point , Solution According to the Quotient Rule, we have So the slope of the tangent line at is This means that the tangeht line at is horizontal and its equation is