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.