The questions in the image provided involve calculating derivatives using rules for differentiation. Let's break down the tasks:

1. Find the derivative of $\csc(x)$ using the derivative proof formula.

We use the known derivative formula for $\csc(x)$: $\frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x)$

2. Find the derivative of $\cot(2x)$ using the derivative proof formula.

For $\cot(2x)$, apply the chain rule: [ \frac{d}{dx}[\cot(2x)] = -\csc^2(2x) \cdot \frac{d}{dx}[2x] = -2\csc^2(2x) ]

3. Calculate the derivatives of the following functions:

(a) $\frac{x^2 \sin(x)}{\sqrt{x^2 - 2x + 5}}$

This is a quotient of two functions, so we apply the quotient rule: $\frac{d}{dx}\left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$ where $f(x) = x^2 \sin(x)$ and $g(x) = \sqrt{x^2 - 2x + 5}$.

(b) $\tan(x) \csc(x)$

Since this is a product of two functions, apply the product rule: $\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$ where $f(x) = \tan(x)$ and $g(x) = \csc(x)$.

(c) $\tan^{-1}(2x)^3 \sqrt{1 - 3x^2}$

This is a product of two functions. Let $u(x) = \tan^{-1}(2x)^3$ and $v(x) = \sqrt{1 - 3x^2}$, and apply the product rule. Then use the chain rule to differentiate each term: $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

(d) $\left( \frac{x^3 + 4}{(1 - 2x^2)^3} \right)^3$

This function involves a chain of expressions. First, differentiate the outer power and then apply the chain rule to differentiate the inner expression: $\frac{d}{dx} \left[ f(x)^3 \right] = 3f(x)^2 \cdot f'(x)$ where $f(x) = \frac{x^3 + 4}{(1 - 2x^2)^3}$. Differentiate $f(x)$ using the quotient rule.

Would you like the detailed steps and solutions for each derivative?

Here are 5 questions related to these concepts:

1. How do you apply the chain rule to trigonometric functions?
2. What is the importance of the quotient rule in calculus?
3. How can you differentiate inverse trigonometric functions?
4. Why is it useful to simplify expressions before differentiating?
5. What are some common mistakes when applying product and chain rules?

Tip: Always ensure you differentiate both the numerator and denominator separately when using the quotient rule.