We are tasked with finding the intervals where the function

$f(x) = \frac{e^x}{1 + e^{4x}}$

is increasing. To do this, we need to calculate the first derivative of $f(x)$ and analyze where it is positive (since a function increases where its derivative is positive).

Step 1: Find the first derivative $f'(x)$

Using the quotient rule:

$f'(x) = \frac{(1 + e^{4x}) \cdot e^x - e^x \cdot 4e^{4x}}{(1 + e^{4x})^2}$

Simplifying the numerator:

$(1 + e^{4x}) e^x - 4e^{4x}e^x = e^x + e^{5x} - 4e^{5x} = e^x - 3e^{5x}$

Thus, the derivative is:

$f'(x) = \frac{e^x (1 - 3e^{4x})}{(1 + e^{4x})^2}$

Step 2: Find the critical points

We need to find where $f'(x) = 0$, which occurs when the numerator equals zero:

$e^x (1 - 3e^{4x}) = 0$

Since $e^x \neq 0$, we solve:

$1 - 3e^{4x} = 0$

$3e^{4x} = 1 \quad \Rightarrow \quad e^{4x} = \frac{1}{3}$

Taking the natural logarithm:

$4x = \ln\left(\frac{1}{3}\right) = -\ln(3) \quad \Rightarrow \quad x = -\frac{1}{4} \ln 3$

Step 3: Determine where $f'(x) > 0$

The function $f(x)$ is increasing where $f'(x) > 0$. From the expression for $f'(x)$:

$f'(x) = \frac{e^x (1 - 3e^{4x})}{(1 + e^{4x})^2}$

We need $1 - 3e^{4x} > 0$, which gives:

$1 > 3e^{4x} \quad \Rightarrow \quad e^{4x} < \frac{1}{3}$

Taking the natural logarithm again:

$4x < \ln\left(\frac{1}{3}\right) = -\ln(3) \quad \Rightarrow \quad x < -\frac{1}{4} \ln 3$

Step 4: Conclusion

The function is increasing for $x < -\frac{1}{4} \ln 3$. Therefore, the interval where $f(x)$ is increasing is:

$\boxed{(-\infty, -\frac{1}{4} \ln 3)}$

This matches option (H) from the choices provided.

Would you like more details on any step? Here are some related questions:

1. How do you apply the quotient rule to find derivatives?
2. What are critical points, and why are they important in determining intervals of increase?
3. How do logarithmic functions help in solving exponential equations?
4. Can you explain how to test intervals for increasing and decreasing behavior using derivatives?
5. Why is it important to analyze the sign of the first derivative?

Tip: Always check critical points and boundary behaviors when analyzing increasing or decreasing functions.