It looks like you've uploaded a mathematical differentiation problem. Let's go over each of them one by one:

1. First Problem: $f(x) = \frac{8x^6}{\sin(x)}$

   To differentiate this, we will apply the quotient rule. The quotient rule states: $\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}$ where $u = 8x^6$ and $v = \sin(x)$. Let's compute their derivatives: $u' = 48x^5, \quad v' = \cos(x)$ Applying the quotient rule: $f'(x) = \frac{48x^5 \sin(x) - 8x^6 \cos(x)}{\sin^2(x)}$

2. Second Problem: $f(x) = \frac{\ln(x)}{7x^2}$

   We will apply the quotient rule here again. Let: $u = \ln(x), \quad v = 7x^2$ Now, the derivatives are: $u' = \frac{1}{x}, \quad v' = 14x$ Applying the quotient rule: $f'(x) = \frac{\frac{1}{x} \cdot 7x^2 - \ln(x) \cdot 14x}{(7x^2)^2}$ Simplifying: $f'(x) = \frac{7x - 14x\ln(x)}{49x^4}$

3. Third Problem: $f(x) = \frac{x^4 + 9x^2 - 1}{x^3}$

   This time, let's first simplify the function by dividing each term in the numerator by $x^3$: $f(x) = x + 9x^{-1} - x^{-3}$ Now, differentiate each term: $f'(x) = 1 - 9x^{-2} + 3x^{-4}$ Therefore: $f'(x) = 1 - \frac{9}{x^2} + \frac{3}{x^4}$

Would you like further clarifications on any of these steps or explanations? Feel free to ask!

Here are some follow-up questions to enhance your understanding:

1. Can you explain the quotient rule for differentiation?
2. How do you handle logarithmic functions when differentiating?
3. What is the chain rule and when would you use it?
4. What are some common mistakes when differentiating trigonometric functions?
5. How do you handle powers of $x$ in differentiation?

Tip: When using the quotient rule, make sure to carefully differentiate both the numerator and denominator and follow through with simplifying the result at the end.