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Let's go through the problems one by one.

### 1. Derivatives of \( f(x) = x^4 - 3x^3 + 2x^2 - 5x + 7 \)
To calculate the first derivative \( f'(x) \):
\[
f'(x) = \frac{d}{dx} \left(x^4 - 3x^3 + 2x^2 - 5x + 7\right)
\]
\[
f'(x) = 4x^3 - 9x^2 + 4x - 5
\]

### 2. Implicit Differentiation of \( x^2 + y^2 = 25 \)
To find \( \frac{dy}{dx} \), differentiate both sides with respect to \( x \):
\[
\frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(25)
\]
\[
2x + 2y \frac{dy}{dx} = 0
\]
\[
\frac{dy}{dx} = -\frac{x}{y}
\]

### 3. Velocity from Position Function \( s(t) = 3t^3 - 6t^2 + 9t - 4 \)
To find the velocity, we need to differentiate the position function \( s(t) \):
\[
v(t) = \frac{ds}{dt} = \frac{d}{dt} \left( 3t^3 - 6t^2 + 9t - 4 \right)
\]
\[
v(t) = 9t^2 - 12t + 9
\]
At \( t = 2 \):
\[
v(2) = 9(2)^2 - 12(2) + 9 = 36 - 24 + 9 = 21
\]
Thus, the velocity at \( t = 2 \) is \( 21 \, \text{m/s} \).

### 4. Chain Rule: \( f(x) = (3x^2 - 4x + 5)^7 \)
Using the chain rule, let \( u = 3x^2 - 4x + 5 \), so:
\[
f'(x) = 7u^6 \cdot \frac{du}{dx}
\]
First, calculate \( \frac{du}{dx} \):
\[
\frac{du}{dx} = 6x - 4
\]
Now substitute back:
\[
f'(x) = 7(3x^2 - 4x + 5)^6 \cdot (6x - 4)
\]

### 5. Second Derivative of \( f(x) = \sin(3x) \)
The first derivative:
\[
f'(x) = \frac{d}{dx} \sin(3x) = 3 \cos(3x)
\]
Now, the second derivative:
\[
f''(x) = \frac{d}{dx} (3 \cos(3x)) = -9 \sin(3x)
\]
The second derivative describes the acceleration or change in the rate of the angular velocity in the context of rotational motion.

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Do you want more details on any specific step or explanation?

Here are 5 related questions for practice:
1. What is the second derivative of \( f(x) = x^4 - 3x^3 + 2x^2 - 5x + 7 \)?
2. If \( x^2 + y^2 = 36 \), what is \( \frac{dy}{dx} \) in terms of \( x \) and \( y \)?
3. What is the acceleration \( a(t) = \frac{dv}{dt} \) for the position function \( s(t) = 3t^3 - 6t^2 + 9t - 4 \)?
4. Find the derivative of \( f(x) = (2x^2 + 3x - 1)^5 \) using the chain rule.
5. What is the third derivative of \( f(x) = \sin(3x) \)?

**Tip:** When working with implicit differentiation, remember to apply the chain rule when differentiating terms involving \( y \), as \( y \) is a function of \( x \).

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Solve a set of calculus problems including first derivatives, implicit differentiation, calculating velocity from position, applying the chain rule, and finding second derivatives. These problems are ideal for university-level calculus students.

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