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.