To find $f'(x)$, the derivative of $f(x) = -\frac{6x^{6/7}}{7} - \frac{5}{x^{-4/3}} + 9$, we first compute the derivative term by term.

Step 1: Rewrite the function for differentiation

Rewrite $f(x)$: $f(x) = -\frac{6}{7} x^{6/7} - 5x^{4/3} + 9$

Step 2: Differentiate each term

1. For the term $-\frac{6}{7}x^{6/7}$, use the power rule $\frac{d}{dx}x^n = n x^{n-1}$: $\frac{d}{dx} \left(-\frac{6}{7}x^{6/7} \right) = -\frac{6}{7} \cdot \frac{6}{7} x^{6/7 - 1} = -\frac{36}{49} x^{-1/7}$

2. For the term $-5x^{4/3}$, use the power rule: $\frac{d}{dx}(-5x^{4/3}) = -5 \cdot \frac{4}{3} x^{4/3 - 1} = -\frac{20}{3} x^{1/3}$

3. For the constant $9$, the derivative is 0: $\frac{d}{dx}(9) = 0$

Step 3: Combine the derivatives

$f'(x) = -\frac{36}{49} x^{-1/7} - \frac{20}{3} x^{1/3}$

Step 4: Evaluate $f'(7)$

Substitute $x = 7$ into the derivative: $f'(7) = -\frac{36}{49} \cdot 7^{-1/7} - \frac{20}{3} \cdot 7^{1/3}$

1. Compute $7^{-1/7}$: $7^{-1/7} = e^{-\ln(7)/7}$

2. Compute $7^{1/3}$: $7^{1/3} = e^{\ln(7)/3}$

These are values that need precise computation, and the exact value requires numerical evaluation.

Numerical Evaluation

If you'd like, I can compute the exact decimal values of these terms. Let me know!

Would you like further clarification, or should I calculate the numerical result?

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Here are five related questions to expand on this topic:

1. How is the power rule applied to functions with fractional exponents?
2. Why is the constant derivative always zero?
3. What are common applications of derivatives in real-world scenarios?
4. How do we find critical points using the derivative?
5. How do we interpret the derivative graphically?

Tip: Always simplify the function as much as possible