To solve the problem and find the derivative of the given function:

$f(x) = \frac{10^x}{10^x + 4}$

We will use the quotient rule for differentiation. The quotient rule states that for a function written as the ratio of two functions, $f(x) = \frac{g(x)}{h(x)}$, the derivative is given by:

$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$

Here, $g(x) = 10^x$ and $h(x) = 10^x + 4$.

Step-by-step solution:

1. Differentiate $g(x) = 10^x$: $g'(x) = \frac{d}{dx} 10^x = 10^x \ln(10)$

2. Differentiate $h(x) = 10^x + 4$: $h'(x) = \frac{d}{dx} (10^x + 4) = 10^x \ln(10)$

3. Apply the quotient rule: $f'(x) = \frac{(10^x \ln(10))(10^x + 4) - (10^x)(10^x \ln(10))}{(10^x + 4)^2}$

4. Simplify the numerator: $f'(x) = \frac{10^x \ln(10) (10^x + 4) - 10^x \cdot 10^x \ln(10)}{(10^x + 4)^2}$ $= \frac{10^x \ln(10) [10^x + 4 - 10^x]}{(10^x + 4)^2}$ $= \frac{10^x \ln(10) \cdot 4}{(10^x + 4)^2}$

5. Final answer: $f'(x) = \frac{4 \cdot 10^x \ln(10)}{(10^x + 4)^2}$

Would you like more details on any of these steps or further clarifications?

Related Questions:

1. What is the quotient rule for differentiation?
2. How does the chain rule apply to exponential functions?
3. How do we differentiate functions of the form $a^x$, where $a$ is a constant?
4. What are common logarithmic identities used in calculus?
5. How do the product and quotient rules differ in application?

Tip: Always remember to apply the natural logarithm when differentiating exponential functions with bases other than $e$.