H(t) = 0.8si [(pi/2)(t-1)] +1.8 find derivative 

To find the derivative of the function \( H(t) = 0.8 \sin\left(\frac{\pi}{2}(t - 1)\right) + 1.8 \), we will apply the chain rule.

1. **Identify the components**: The function consists of a sine function and a constant.

2. **Differentiate the sine function**:
   - The derivative of \( \sin(x) \) is \( \cos(x) \).
   - In our case, \( x = \frac{\pi}{2}(t - 1) \).

3. **Apply the chain rule**:
   - If \( u = \frac{\pi}{2}(t - 1) \), then the derivative \( \frac{du}{dt} = \frac{\pi}{2} \).
   - Therefore, the derivative of \( \sin(u) \) is \( \cos(u) \cdot \frac{du}{dt} = \cos\left(\frac{\pi}{2}(t - 1)\right) \cdot \frac{\pi}{2} \).

4. **Combine everything**:
   - The derivative \( H'(t) \) is:
   \[
   H'(t) = 0.8 \cdot \cos\left(\frac{\pi}{2}(t - 1)\right) \cdot \frac{\pi}{2}
   \]
   - Since the constant \( 1.8 \) does not contribute to the derivative, it is omitted.

5. **Final expression**:
   \[
   H'(t) = 0.4\pi \cos\left(\frac{\pi}{2}(t - 1)\right)
   \]

Would you like more details on any step, or do you have further questions? Here are some related questions you might find interesting:

1. What is the value of \( H'(t) \) at \( t = 1 \)?
2. How would you find the second derivative \( H''(t) \)?
3. Can you explain the significance of the sine and cosine functions in this context?
4. What are the applications of this type of function in real-world scenarios?
5. How would you graph \( H(t) \) and its derivative \( H'(t) \)? 

**Tip**: Always remember to check the function's domain and behavior when working with trigonometric derivatives!

H(t) = 0.8si [(pi/2)(t-1)] +1.8 find derivative

Explore how to find the derivative of the function H(t) = 0.8sin((π/2)(t-1)) + 1.8 using the chain rule and trigonometric identities. This detailed solution is ideal for high school students in grades 11-12.

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