To differentiate the function $f(x) = \ln^2(x) - 4\ln(x) + 3$, we will use the chain rule and basic derivative rules.

Step-by-step differentiation:

1. Differentiate $\ln^2(x)$: $\frac{d}{dx} \ln^2(x) = 2\ln(x) \cdot \frac{d}{dx}[\ln(x)] = 2\ln(x) \cdot \frac{1}{x} = \frac{2\ln(x)}{x}$

2. Differentiate $-4\ln(x)$: [ \frac{d}{dx}[-4\ln(x)] = -4 \cdot \frac{d}{dx}[\ln(x)] = -4 \cdot \frac{1}{x} = -\frac{4}{x} ]

3. Differentiate the constant $3$: $\frac{d}{dx}[3] = 0$

Putting it all together:

$f'(x) = \frac{2\ln(x)}{x} - \frac{4}{x}$

Thus, the derivative of the function is: $f'(x) = \frac{2\ln(x) - 4}{x}$

Would you like further clarification or details on any step?

Relative Questions:

1. What is the chain rule, and how is it applied in differentiation?
2. How do you find the derivative of logarithmic functions?
3. What are the properties of natural logarithms useful for calculus?
4. Can you differentiate functions involving higher powers of logarithms?
5. How would the derivative change if the function included other logarithmic bases?

Tip:

Always simplify expressions after differentiating, especially when dealing with logarithms, to make calculations easier!