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Let's evaluate the derivatives of each given expression.

### 1. \( \frac{d}{dx} \left( x^9 + x^8 + x^7 + 5 \right) \)

- Derivative of \( x^9 \) is \( 9x^8 \)
- Derivative of \( x^8 \) is \( 8x^7 \)
- Derivative of \( x^7 \) is \( 7x^6 \)
- Derivative of the constant \( 5 \) is \( 0 \)

Thus, 
\[
\frac{d}{dx} \left( x^9 + x^8 + x^7 + 5 \right) = 9x^8 + 8x^7 + 7x^6
\]

### 2. \( \frac{d}{dx} \left( 15x^{28} - 36x^{44} - 56x^{63} \right) \)

- Derivative of \( 15x^{28} \) is \( 15 \times 28x^{27} = 420x^{27} \)
- Derivative of \( -36x^{44} \) is \( -36 \times 44x^{43} = -1584x^{43} \)
- Derivative of \( -56x^{63} \) is \( -56 \times 63x^{62} = -3528x^{62} \)

Thus,
\[
\frac{d}{dx} \left( 15x^{28} - 36x^{44} - 56x^{63} \right) = 420x^{27} - 1584x^{43} - 3528x^{62}
\]

### 3. \( \frac{d}{dx} \left( 8x^{-1} - 8e^x \right) \)

- Derivative of \( 8x^{-1} \) is \( 8 \times (-1)x^{-2} = -8x^{-2} \)
- Derivative of \( -8e^x \) is \( -8e^x \), because \( e^x \) remains the same during differentiation.

Thus,
\[
\frac{d}{dx} \left( 8x^{-1} - 8e^x \right) = -8x^{-2} - 8e^x
\]

### 4. \( \frac{d}{dx} \left( 7x^{4.3} - 9x^{0.1} + x^\pi \right) \)

- Derivative of \( 7x^{4.3} \) is \( 7 \times 4.3x^{3.3} = 30.1x^{3.3} \)
- Derivative of \( -9x^{0.1} \) is \( -9 \times 0.1x^{-0.9} = -0.9x^{-0.9} \)
- Derivative of \( x^\pi \) is \( \pi x^{\pi - 1} \), as \( \pi \) is a constant exponent.

Thus,
\[
\frac{d}{dx} \left( 7x^{4.3} - 9x^{0.1} + x^\pi \right) = 30.1x^{3.3} - 0.9x^{-0.9} + \pi x^{\pi - 1}
\]

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Would you like a deeper explanation on any of the steps?

Here are 5 related questions you might find interesting:
1. How do you differentiate logarithmic functions like \( \ln(x) \)?
2. How do you compute the derivative of a product of two functions?
3. What is the chain rule in differentiation?
4. How do you handle higher-order derivatives like the second or third derivative?
5. What is the derivative of a trigonometric function like \( \sin(x) \)?

**Tip:** Always simplify expressions where possible before taking the derivative to avoid unnecessary complexity!

Evaluate the following derivatives: 
1. d/dx [x^9 + x^8 + x^7 + 5] 
2. d/dx [15x^28 - 36x^44 - 56x^63] 
3. d/dx [8x^(-1) - 8e^x] 
4. d/dx [7x^4.3 - 9x^0.1 + x^π]

Learn how to evaluate derivatives of complex polynomial and exponential functions, covering key differentiation rules like the Power Rule and the derivative of exponential functions. This problem is suitable for high school to college-level students.

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