The problem presented is asking for a derivative calculation using symbolic rules and a numeric approximation. Specifically, the function given is $h(x) = 1.22^x$, and the task is to find the derivative of this function at $x = 5$.

Step 1: Symbolic Derivative

The derivative of $h(x) = 1.22^x$ can be found using the chain rule. The derivative of $a^x$ (where $a$ is a constant) is given by: $\frac{d}{dx} a^x = a^x \ln(a)$ Applying this rule to $h(x) = 1.22^x$: $h'(x) = 1.22^x \ln(1.22)$

At $x = 5$, the derivative becomes: $h'(5) = 1.22^5 \ln(1.22)$

Step 2: Numeric Evaluation

Now let's calculate the numeric value for this:

• $1.22^5 \approx 2.48832$
• $\ln(1.22) \approx 0.19867$

Thus: $h'(5) \approx 2.48832 \times 0.19867 \approx 0.4945$

Step 3: Numerical Approximation Using the "Calculator Formula"

To approximate the derivative numerically, we can use the finite difference approximation formula: $h'(x) \approx \frac{h(x+h) - h(x)}{h}$ where $h$ is a small value, typically something like $h = 0.001$.

Using Excel, you would compute: $h'(5) \approx \frac{h(5.001) - h(5)}{0.001}$

Given the results from the problem, the numeric and symbolic derivatives agree to 1 decimal digit.

Would you like further details on any step, or help calculating the exact numeric derivative using a specific step size? Here are a few follow-up questions:

1. What is the significance of choosing a small $h$ for the finite difference formula?
2. How would increasing the precision of the finite difference method affect the results?
3. Can you derive the exact form of the error in using a finite difference approximation?
4. How does the chain rule help in differentiating functions like $1.22^x$?
5. What are other methods for numerically approximating derivatives?

Tip: When using Excel for numeric derivatives, ensure that your chosen step size $h$ is small enough to reduce error but not too small to introduce round-off errors.