Let's solve problems 9 and 10 step by step.

Problem 9:

$\frac{d}{dx} \left( \frac{4x^2}{\ln x} \right)$ This is a quotient rule problem. The quotient rule states: $\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ where:

• $f(x) = 4x^2$
• $g(x) = \ln x$

Step 1: Differentiate $f(x) = 4x^2$ $f'(x) = 8x$

Step 2: Differentiate $g(x) = \ln x$ $g'(x) = \frac{1}{x}$

Step 3: Apply quotient rule $\frac{(8x \cdot \ln x) - (4x^2 \cdot \frac{1}{x})}{(\ln x)^2}$ $= \frac{8x \ln x - 4x}{(\ln x)^2}$

Final Answer:
$\frac{8x \ln x - 4x}{(\ln x)^2}$

Problem 10:

$\frac{d}{dx} \left( \frac{3 \sin x}{\tan x} \right)$ This also follows the quotient rule, where:

• $f(x) = 3\sin x$
• $g(x) = \tan x$

Step 1: Differentiate $f(x) = 3\sin x$ $f'(x) = 3\cos x$

Step 2: Differentiate $g(x) = \tan x$ $g'(x) = \sec^2 x$

Step 3: Apply quotient rule $\frac{(3\cos x \cdot \tan x) - (3\sin x \cdot \sec^2 x)}{(\tan x)^2}$

Final Answer:
$\frac{3\cos x \tan x - 3\sin x \sec^2 x}{\tan^2 x}$

Would you like more explanation on any step?

Here are 5 related problems you can try:

1. $\frac{d}{dx} \left( \frac{x^2}{e^x} \right)$
2. $\frac{d}{dx} \left( \frac{x^3}{\ln x} \right)$
3. $\frac{d}{dx} \left( \frac{e^x}{x} \right)$
4. $\frac{d}{dx} \left( x^x \right)$
5. $\frac{d}{dx} \left( \frac{\sin x}{x^2} \right)$

Tip:
For derivatives of fractions, always check if the quotient rule is required or if simplification can be done first.