Find the derivative of the function.
f(x) = 16ex + 19^x

To find the derivative of the function \( f(x) = 16e^x + 19^x \), we'll differentiate each term separately.

1. **Differentiate \( 16e^x \):**
   \[
   \frac{d}{dx}\left(16e^x\right) = 16 \cdot \frac{d}{dx}\left(e^x\right) = 16e^x
   \]

2. **Differentiate \( 19^x \):**
   To differentiate \( 19^x \), we use the formula for the derivative of an exponential function \( a^x \), which is \( \frac{d}{dx}\left(a^x\right) = a^x \ln(a) \). Here, \( a = 19 \).
   \[
   \frac{d}{dx}\left(19^x\right) = 19^x \ln(19)
   \]

Now, combining these results:

\[
f'(x) = 16e^x + 19^x \ln(19)
\]

This is the derivative of the function \( f(x) \).

Would you like more details or have any questions?

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Here are some related questions to further your understanding:

1. How would the derivative change if the base of the exponential function \( 19^x \) were different?
2. What is the derivative of the function \( g(x) = a^x + b^x \) for arbitrary constants \( a \) and \( b \)?
3. How does the derivative of a function with a base \( e \) differ from that with a general base?
4. What is the second derivative of \( f(x) = 16e^x + 19^x \)?
5. How do logarithmic functions relate to derivatives of exponential functions?

**Tip:** When differentiating functions with constant bases, remember that the derivative involves multiplying by the natural logarithm of the base.

Learn how to find the derivative of the function f(x) = 16e^x + 19^x step-by-step. This problem involves differentiation of exponential functions and the use of the derivative formula. Suitable for advanced high school students and calculus enthusiasts.

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