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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 \( e \).

Find the derivative of f(x) = 10^x / (10^x + 4).

Learn how to find the derivative of f(x) = 10^x / (10^x + 4) using the quotient rule. This step-by-step solution covers key calculus concepts like differentiation, the quotient rule, and exponential function properties. Suitable for high school calculus students.

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