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The image contains 8 mathematical expressions where each \( f(x) \) appears to be a function involving different algebraic, exponential, logarithmic, or trigonometric forms. I'll provide solutions by differentiating each function \( f(x) \).

### (1) \( f(x) = 7^{3x^3 - x^2 + 3x} \)

To differentiate this, let's denote:
\[
y = 7^{3x^3 - x^2 + 3x}
\]
Taking the natural logarithm on both sides, we get:
\[
\ln y = (3x^3 - x^2 + 3x) \ln 7
\]
Now, differentiate implicitly:
\[
\frac{1}{y} \frac{dy}{dx} = (9x^2 - 2x + 3) \ln 7
\]
Thus:
\[
f'(x) = y(9x^2 - 2x + 3) \ln 7 = 7^{3x^3 - x^2 + 3x}(9x^2 - 2x + 3) \ln 7
\]

### (2) \( f(x) = e^{5x^3 - 3x^2} \)

To differentiate:
\[
f'(x) = \frac{d}{dx} \left( e^{5x^3 - 3x^2} \right) = e^{5x^3 - 3x^2} \cdot (15x^2 - 6x)
\]
\[
f'(x) = (15x^2 - 6x) e^{5x^3 - 3x^2}
\]

### (3) \( f(x) = e^{\sqrt{2x + 1}} \)

Let \( u = \sqrt{2x + 1} \), then:
\[
f'(x) = e^{\sqrt{2x + 1}} \cdot \frac{d}{dx}(\sqrt{2x + 1})
\]
\[
= e^{\sqrt{2x + 1}} \cdot \frac{1}{2\sqrt{2x + 1}} \cdot 2 = \frac{e^{\sqrt{2x + 1}}}{\sqrt{2x + 1}}
\]

### (4) \( f(x) = 4 \log(2x^5 - 3x^3 + x) \)

Using the chain rule:
\[
f'(x) = 4 \cdot \frac{1}{2x^5 - 3x^3 + x} \cdot (10x^4 - 9x^2 + 1)
\]
\[
f'(x) = \frac{4(10x^4 - 9x^2 + 1)}{2x^5 - 3x^3 + x}
\]

### (5) \( f(x) = (5x^2 + 3x) \log(7x + 3) \)

Let \( u = 5x^2 + 3x \) and \( v = \log(7x + 3) \). Using the product rule:
\[
f'(x) = u'v + uv' = (10x + 3) \log(7x + 3) + (5x^2 + 3x) \frac{7}{7x + 3}
\]

### (6) \( f(x) = (2x^3 - 4x) e^{2x^5} \)

Using the product rule:
\[
f'(x) = (6x^2 - 4) e^{2x^5} + (2x^3 - 4x) \cdot e^{2x^5} \cdot 10x^4
\]

### (7) \( f(x) = \sin(4x^3 - 2x) \)

\[
f'(x) = \cos(4x^3 - 2x) \cdot (12x^2 - 2)
\]

### (8) \( f(x) = \ln\left(\frac{1}{\sqrt[5]{x^2 + 3x}}\right) \)

Simplify first:
\[
f(x) = -\frac{1}{5} \ln(x^2 + 3x)
\]
Differentiate:
\[
f'(x) = -\frac{1}{5} \cdot \frac{2x + 3}{x^2 + 3x}
\]

These are the derivatives for each problem. Would you like further details or explanations on any specific function? 

### Follow-up Questions:
1. Can you simplify \( f'(x) \) further in example (6)?
2. How would you find the second derivative of \( f(x) \) in problem (7)?
3. What is the interpretation of the derivative result in problem (5)?
4. Could \( f'(x) \) be zero for any \( x \) in the given functions?
5. What would happen if you changed the base of the logarithmic functions in problems (4) and (5)?

### Tip:
Always simplify the function as much as possible before differentiating; this can often make the process easier and reduce errors.

Differentiate the given mathematical functions.

Step-by-step differentiation of various functions, including exponential, logarithmic, and trigonometric expressions. Covers the product rule, chain rule, and related calculus principles for Grade 11-12 students.

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