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We are given the function for population change:

\[
P(t) = 5.4^t
\]

### Let's analyze the provided statements.

#### (i) \( P'(t) = \lim_{h \to 0} \frac{5.4^h(5.4^t - 1)}{h} \)
This expression represents the derivative of \( P(t) \), essentially \( P'(t) \), which can be calculated using logarithmic differentiation or applying the chain rule for the exponential function.

#### (ii) \( P'(1) \) is less than the average rate of change of \( P \) between \( t = 0 \) and \( t = 1 \).
Let's compute the instantaneous rate of change \( P'(1) \) by differentiating \( P(t) = 5.4^t \):

\[
P'(t) = 5.4^t \ln(5.4)
\]

Thus,

\[
P'(1) = 5.4^1 \ln(5.4) = 5.4 \ln(5.4)
\]

The average rate of change between \( t = 0 \) and \( t = 1 \) is:

\[
\text{Average Rate of Change} = \frac{P(1) - P(0)}{1 - 0} = \frac{5.4^1 - 5.4^0}{1} = 5.4 - 1 = 4.4
\]

Now, compare \( P'(1) \) and 4.4:

\[
P'(1) = 5.4 \times \ln(5.4) \approx 5.4 \times 1.686 = 9.11
\]

Since \( 9.11 > 4.4 \), statement (ii) is **false**.

#### (iii) The average rate of change between 0 and 1 is 5.4.
We already computed the average rate of change between 0 and 1:

\[
\frac{P(1) - P(0)}{1 - 0} = 4.4
\]

Thus, statement (iii) is **false** as well.

### Conclusion:
- Statement (i) involves correct differentiation but we must check its comparison to the others.
- Statement (ii) is **false**.
- Statement (iii) is **false**.

Thus, the correct answer is:

\[
\boxed{D} \text{ (none of them)}
\]

Do you need more details, or would you like further clarification?

### Related Questions:
1. What is the derivative of exponential functions in general?
2. How can you find the average rate of change between two points?
3. What is the difference between instantaneous and average rates of change?
4. How does logarithmic differentiation help in solving exponential derivatives?
5. Why is the logarithm of a number important in calculating derivatives of exponential functions?

**Tip:** When comparing instantaneous and average rates of change, the derivative gives you the rate at a specific point, while the average rate gives you the overall change between two points.

A population changes according to the formula P(t) = 5.4^t. Which of the following statements are always true? (i) P'(t) = lim h→0 5.4^h(5.4^t−1)/h (ii) P'(1) is less than the average rate of change of P between 0 and 1. (iii) The average rate of change between 0 and 1 is 5.4.

Explore the population change function P(t) = 5.4^t. Learn how to calculate the derivative P'(t) and the average rate of change between t=0 and t=1, involving key concepts like exponential functions and logarithmic differentiation. Suitable for college-level calculus students.

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