We want to find the derivative of the function:

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

We will apply the quotient rule to differentiate this function. The quotient rule states:

$\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2}$

Here, let:

• $u(x) = 10^x$
• $v(x) = 10^x + 4$

Step 1: Differentiate $u(x)$ and $v(x)$

• The derivative of $u(x) = 10^x$ is: $u'(x) = 10^x \ln(10)$

• The derivative of $v(x) = 10^x + 4$ is: $v'(x) = 10^x \ln(10)$

Step 2: Apply the quotient rule

Now, applying the quotient rule:

$f'(x) = \frac{(10^x \ln(10)) \cdot (10^x + 4) - 10^x \cdot (10^x \ln(10))}{(10^x + 4)^2}$

Step 3: Simplify the expression

Simplifying the numerator:

$f'(x) = \frac{10^x \ln(10) \cdot (10^x + 4) - 10^{2x} \ln(10)}{(10^x + 4)^2}$

Distribute $10^x \ln(10)$ inside the parentheses:

$f'(x) = \frac{10^{2x} \ln(10) + 4 \cdot 10^x \ln(10) - 10^{2x} \ln(10)}{(10^x + 4)^2}$

Notice that the terms $10^{2x} \ln(10)$ cancel each other out:

$f'(x) = \frac{4 \cdot 10^x \ln(10)}{(10^x + 4)^2}$

Thus, the derivative of the function is:

$f'(x) = \frac{4 \cdot 10^x \ln(10)}{(10^x + 4)^2}$

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Would you like a more detailed explanation of any of the steps, or do you have any further questions?

Here are some related questions you might find interesting:

1. What is the derivative of $\frac{a^x}{a^x + b}$?
2. How do you apply the chain rule in differentiation?
3. What is the behavior of the derivative as $x \to \infty$?
4. How do logarithmic properties help in differentiating exponential functions?
5. What is the second derivative of this function?

Tip: When applying the quotient rule, always simplify the result as much as possible before moving on to the next step.