Let's solve each problem step-by-step.

1. Find the derivative of $y = \ln(x - 3)$:

The derivative of $\ln(u)$ is $\frac{1}{u} \cdot \frac{du}{dx}$.

Here:

• $u = x - 3$
• $\frac{du}{dx} = 1$

Thus: $\frac{dy}{dx} = \frac{1}{x - 3} \cdot 1 = \frac{1}{x - 3}.$

Answer: D) $\frac{1}{x - 3}$.

2. Find the derivative of $y = \frac{\ln x}{x^6}$:

Use the quotient rule: $\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2}.$ Here:

• $u = \ln x$, so $u' = \frac{1}{x}$,
• $v = x^6$, so $v' = 6x^5$.

Now compute: $\frac{dy}{dx} = \frac{\left( \frac{1}{x} \cdot x^6 \right) - \left( \ln x \cdot 6x^5 \right)}{(x^6)^2}.$

Simplify: $\frac{dy}{dx} = \frac{x^5 - 6x^5 \ln x}{x^{12}} = \frac{1 - 6 \ln x}{x^7}.$

Answer: C) $\frac{1 - 6 \ln x}{x^7}$.

3. Find the derivative of $y = (x^2 - 2x + 4)e^x$:

Use the product rule: $\frac{d}{dx}(uv) = u'v + uv',$ where $u = (x^2 - 2x + 4)$ and $v = e^x$.

• $u = x^2 - 2x + 4$, so $u' = 2x - 2$,
• $v = e^x$, so $v' = e^x$.

Now compute: $\frac{dy}{dx} = (2x - 2)e^x + (x^2 - 2x + 4)e^x.$

Factor out $e^x$: $\frac{dy}{dx} = \left( (2x - 2) + (x^2 - 2x + 4) \right)e^x.$

Simplify the terms inside the parentheses: $(2x - 2) + (x^2 - 2x + 4) = x^2 + 4.$

Thus: $\frac{dy}{dx} = (x^2 + 2)e^x.$

Answer: C) $(x^2 + 2)e^x$.

4. Find the derivative of $y = 3 \sin^{-1}(5x^4)$:

Use the chain rule for $\sin^{-1}(u)$: $\frac{d}{dx}(\sin^{-1}(u)) = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx}.$

Here:

• $u = 5x^4$, so $\frac{du}{dx} = 20x^3$.

Now compute: $\frac{dy}{dx} = 3 \cdot \frac{1}{\sqrt{1 - (5x^4)^2}} \cdot 20x^3.$

Simplify: $\frac{dy}{dx} = \frac{60x^3}{\sqrt{1 - 25x^8}}.$

Answer: B) $\frac{60x^3}{\sqrt{1 - 25x^8}}$.

Final Answers:

1. D) $\frac{1}{x - 3}$
2. C) $\frac{1 - 6 \ln x}{x^7}$
3. C) $(x^2 + 2)e^x$
4. B) $\frac{60x^3}{\sqrt{1 - 25x^8}}$

Would you like me to explain any solution in more detail? Let me know if you need help with another step.

Here are 5 follow-up questions to test your understanding:

1. What is the derivative of $y = \ln(x + 5)$?
2. How do you differentiate $y = x^3e^x$ using the product rule?
3. Find the derivative of $y = \frac{1}{x^4}$.
4. What is the derivative of $y = \arcsin(2x)$?
5. Can you apply the quotient rule to differentiate $y = \frac{x^2}{x^3 + 1}$?

Tip: Always look for opportunities to simplify expressions before applying derivative rules!