To estimate the derivative of $f(x) = 1.3^{1.9x}$ at $x = 2.1$, we can use the definition of the derivative:

$f'(x) \approx \frac{f(x+h) - f(x)}{h}$ for a small value of $h$.

We'll choose a small $h$ (for example, $h = 0.001$) and compute:

$f'(2.1) \approx \frac{f(2.1 + h) - f(2.1)}{h}$

Step 1: Calculate $f(2.1)$

$f(2.1) = 1.3^{1.9 \times 2.1} = 1.3^{3.99}$

Step 2: Calculate $f(2.1 + h)$ for $h = 0.001$

$f(2.1 + 0.001) = 1.3^{1.9 \times 2.101} = 1.3^{3.9919}$

Step 3: Apply the formula

Now I'll calculate these values and estimate the derivative.The estimated derivative $f'(2.1)$ is approximately $1.42$.

Would you like more details on the steps, or do you have any questions?

Here are 5 related questions to expand on this concept:

1. How does the choice of $h$ affect the accuracy of the derivative estimate?
2. Can we use a different method, like symbolic differentiation, to find the exact derivative of $f(x)$?
3. How would the process change if $f(x)$ was a trigonometric or logarithmic function?
4. What is the geometric interpretation of the derivative $f'(x)$ at a given point?
5. Can you approximate the second derivative $f''(x)$ using a similar method?

Tip: Smaller values of $h$ give more accurate derivative approximations, but extremely small values might introduce numerical errors due to floating-point precision limits.