We are tasked with finding the derivative of the function

$f(x) = \frac{e^{-x} \sec x}{1 + e^x}.$

Let’s proceed step by step:

---

Step 1: Rewrite the function

The function $f(x)$ is already in the quotient form:

$f(x) = \frac{u(x)}{v(x)},$

where $u(x) = e^{-x} \sec x, \quad v(x) = 1 + e^x.$

---

Step 2: Apply the Quotient Rule

The derivative of a quotient $\frac{u(x)}{v(x)}$ is given by:

$f'(x) = \frac{v(x) u'(x) - u(x) v'(x)}{(v(x))^2}.$

---

Step 3: Compute $u'(x)$ and $v'(x)$

$u(x) = e^{-x} \sec x$

Using the product rule for derivatives: $u'(x) = \frac{d}{dx} \left(e^{-x} \sec x \right) = \frac{d}{dx} \left(e^{-x}\right) \sec x + e^{-x} \frac{d}{dx} \left(\sec x\right).$

1. Derivative of $e^{-x}$: $\frac{d}{dx} \left(e^{-x}\right) = -e^{-x}.$

2. Derivative of $\sec x$: $\frac{d}{dx} \left(\sec x\right) = \sec x \tan x.$

So: $u'(x) = \left(-e^{-x}\right) \sec x + e^{-x} \sec x \tan x = e^{-x} \sec x (\tan x - 1).$

$v(x) = 1 + e^x$

$v'(x) = \frac{d}{dx} \left(1 + e^x\right) = 0 + e^x = e^x.$

---

Step 4: Substitute into the Quotient Rule

Now substitute $u'(x)$, $u(x)$, $v'(x)$, and $v(x)$ into the quotient rule:

$f'(x) = \frac{(1 + e^x)\left(e^{-x} \sec x (\tan x - 1)\right) - \left(e^{-x} \sec x\right) \left(e^x\right)}{\left(1 + e^x\right)^2}.$

---

Step 5: Simplify the numerator

Distribute the terms in the numerator:

1. First term: $(1 + e^x)\left(e^{-x} \sec x (\tan x - 1)\right) = e^{-x} \sec x (\tan x - 1) + e^x \cdot e^{-x} \sec x (\tan x - 1).$

Simplify: $e^{-x} \sec x (\tan x - 1) + \sec x (\tan x - 1).$

2. Second term: $-e^{-x} \sec x e^x = -\sec x.$

Combine the two terms: $e^{-x} \sec x (\tan x - 1) + \sec x (\tan x - 1) - \sec x.$

---

Final Answer:

$f'(x) = \frac{e^{-x} \sec x (\tan x - 1) + \sec x (\tan x - 2)}{(1 + e^x)^2}.$

This is the derivative of $f(x)$.

---

Let me know if you'd like further details or a specific part clarified! 😊

---

Related Questions:

1. How do you differentiate functions involving exponential and trigonometric terms?
2. What is the product rule, and when do you use it?
3. How do you apply the quotient rule step by step for complex functions?
4. Why is it necessary to simplify after differentiation?
5. Can you explain the derivative of $\sec x$ in more detail?

Tip:

For complicated derivatives, work on the numerator and denominator separately before applying the rules—it keeps calculations cleaner!