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The image shows three problems involving derivatives and logarithms. Let's solve them one by one.

### Problem 1:
Given:
\[
y = \log_2(2x^3 + 3x - 4)
\]

To differentiate \( y \) with respect to \( x \), we will use the chain rule. Recall the derivative of a logarithmic function:
\[
\frac{d}{dx} \log_a(u) = \frac{1}{u \ln(a)} \cdot \frac{du}{dx}
\]
where \( u = 2x^3 + 3x - 4 \) and \( a = 2 \).

First, let's find \( \frac{du}{dx} \):
\[
\frac{d}{dx} (2x^3 + 3x - 4) = 6x^2 + 3
\]

Now, using the formula for the derivative of a logarithm:
\[
\frac{dy}{dx} = \frac{1}{(2x^3 + 3x - 4) \ln(2)} \cdot (6x^2 + 3)
\]

Thus, the derivative is:
\[
\frac{dy}{dx} = \frac{6x^2 + 3}{(2x^3 + 3x - 4) \ln(2)}
\]

---

### Problem 2:
Given:
\[
f(t) = \ln(\ln(t))
\]

We apply the chain rule again. The derivative of \( \ln(u) \) is \( \frac{1}{u} \), and we have \( u = \ln(t) \).

First, differentiate the outer function:
\[
\frac{d}{dt} \ln(\ln(t)) = \frac{1}{\ln(t)}
\]

Now, multiply by the derivative of the inner function \( \ln(t) \):
\[
\frac{d}{dt} \ln(t) = \frac{1}{t}
\]

Thus, the derivative is:
\[
f'(t) = \frac{1}{t \ln(t)}
\]

---

### Problem 3:
Given:
\[
g(x) = \left( \log_7(x^2 + x + 1) \right)^5
\]

This is a composite function, so we apply the chain rule. First, we differentiate the outer function \( u^5 \) with respect to \( u = \log_7(x^2 + x + 1) \), then differentiate the inner function.

First, the outer function:
\[
\frac{d}{du} u^5 = 5u^4
\]

Now, apply the chain rule:
\[
g'(x) = 5 \left( \log_7(x^2 + x + 1) \right)^4 \cdot \frac{d}{dx} \log_7(x^2 + x + 1)
\]

We now differentiate the logarithm \( \log_7(x^2 + x + 1) \):
\[
\frac{d}{dx} \log_a(u) = \frac{1}{u \ln(a)} \cdot \frac{du}{dx}
\]
where \( u = x^2 + x + 1 \) and \( a = 7 \).

First, compute \( \frac{du}{dx} \):
\[
\frac{d}{dx} (x^2 + x + 1) = 2x + 1
\]

Thus:
\[
\frac{d}{dx} \log_7(x^2 + x + 1) = \frac{2x + 1}{(x^2 + x + 1) \ln(7)}
\]

Now substitute back into the expression for \( g'(x) \):
\[
g'(x) = 5 \left( \log_7(x^2 + x + 1) \right)^4 \cdot \frac{2x + 1}{(x^2 + x + 1) \ln(7)}
\]

---

Do you need further clarification on any of these solutions? Let me know if you'd like a detailed breakdown of any steps!

Here are five related questions you might want to explore:
1. How do you differentiate \( \log_b(f(x)) \) when \( f(x) \) is more complex?
2. Can you apply the product rule and chain rule together in logarithmic functions?
3. How would you handle logarithmic derivatives when the base is a variable rather than a constant?
4. What are the applications of logarithmic differentiation in real-world problems?
5. How would the rules change if natural logarithms were involved?

**Tip:** Remember that logarithmic differentiation is especially useful when dealing with products or quotients of functions, particularly when those functions are raised to powers.

Compute the derivatives of the following functions: y = log2(2x^3 + 3x - 4), f(t) = ln(ln(t)), g(x) = (log7(x^2 + x + 1))^5.

Explore the step-by-step solutions to derivative problems involving logarithmic functions and the chain rule. Problems include log base 2, natural logarithms, and composite functions, suitable for advanced calculus students.

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