To solve these problems, let's go through each part of the question and outline the steps needed to compute the derivatives. Here’s a breakdown of each sub-question:

Problem 3: Compute the derivatives of the following functions.

Part (a): Fractional Linear Transformation

For $f(x) = \frac{ax + b}{cx + d}$, where $a, b, c, d$ are constants and $ad - bc = 1$:

1. Use the quotient rule for derivatives: If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'v - uv'}{v^2}$.
2. Set $u(x) = ax + b$ and $v(x) = cx + d$, and find $u'(x)$ and $v'(x)$.
3. Substitute into the quotient rule formula to find $f'(x)$.

Part (b): Polynomials (factored in terms of their zeroes)

For $P(x) = \prod_{k=1}^n (x - a_k)$, where $a_1, \dots, a_n$ are constants:

1. This is a product of linear terms, so apply the product rule iteratively.
2. Alternatively, note that for a polynomial in factored form, each term’s derivative is $\frac{d}{dx}(x - a_i) = 1$.
3. Use the fact that $P'(x)$ is a sum of terms where each $(x - a_i)$ is differentiated once.

Part (c): $f(x) = \cos(x + x^2)$

1. Apply the chain rule.
2. Let $u = x + x^2$, so $f(x) = \cos(u)$.
3. Find $f'(x) = -\sin(u) \cdot u'(x)$.
4. Differentiate $u(x) = x + x^2$ to get $u'(x)$.

Part (d): $\sin^3(\sin^2(\sin(x)))$

1. This involves a composition of multiple functions, so apply the chain rule multiple times.
2. Let $u = \sin(x)$, $v = \sin^2(u)$, and $w = \sin^3(v)$.
3. Differentiate $w$ with respect to $v$, $v$ with respect to $u$, and $u$ with respect to $x$, and then multiply these derivatives.

Part (e): $\sin\left( \frac{x - \sin\left(\frac{x}{x - \sin(x)}\right)}{x - \sin\left(\frac{x}{x - \sin(x)}\right)} \right)$

1. This is a highly nested expression, so proceed with the chain rule and quotient rule as necessary.
2. Define inner functions and differentiate layer by layer, working from the innermost terms outward.

Would you like me to work through the derivatives step-by-step for each part, or is there a specific part you'd like to focus on?

Related Questions

1. How do you apply the chain rule for functions with multiple layers of composition?
2. What is the quotient rule, and how does it apply to fractional linear transformations?
3. How can the product rule be used for polynomials factored in terms of their zeroes?
4. How does one differentiate nested trigonometric functions?
5. What techniques are best for managing complex, multi-layered trigonometric derivatives?

Tip:

When faced with complex expressions, break them down into smaller functions and apply differentiation rules step-by-step, moving from the innermost function outward.