I see you've uploaded a photo with a list of functions to differentiate. Let's go over each one step by step.

1. $y = \ln(3x + 1)$

Using the chain rule for derivatives: $\frac{dy}{dx} = \frac{1}{3x + 1} \cdot 3 = \frac{3}{3x + 1}$

2. $y = 4\ln(x^2)$

Apply the logarithmic property $\ln(a^b) = b\ln(a)$: $y = 4 \cdot 2\ln(x) = 8\ln(x)$ Now differentiate: $\frac{dy}{dx} = 8 \cdot \frac{1}{x} = \frac{8}{x}$

3. $y = e^{5x - 3}$

For the derivative of $e^{u(x)}$, we use $\frac{d}{dx} e^{u(x)} = e^{u(x)} \cdot u'(x)$: $\frac{dy}{dx} = e^{5x - 3} \cdot 5 = 5e^{5x - 3}$

4. $y = x e^{4x + 1}$

This requires the product rule $\frac{d}{dx} [u(x) v(x)] = u'(x)v(x) + u(x)v'(x)$: Let $u = x$ and $v = e^{4x + 1}$: $u' = 1, \quad v' = e^{4x + 1} \cdot 4$ Now apply the product rule: $\frac{dy}{dx} = 1 \cdot e^{4x + 1} + x \cdot 4e^{4x + 1} = e^{4x + 1} + 4x e^{4x + 1}$

5. $y = 5^{2x}$

Use the chain rule and derivative of exponential functions: $\frac{dy}{dx} = 5^{2x} \cdot \ln(5) \cdot 2 = 2 \ln(5) \cdot 5^{2x}$

Would you like me to continue with the other functions as well? Let me know if you'd like further explanations or details.

Related Questions:

1. What is the derivative of a function involving logarithms?
2. How does the chain rule work in differentiation?
3. Can you explain the product rule in more detail?
4. What happens when you differentiate a function with a base other than $e$?
5. How do you handle derivatives of exponential functions with constant bases?

Tip: When applying the chain rule, remember that you must always differentiate the inner function as well as the outer function.