To solve the given problem, we are asked to find the derivative of $f(x) = \frac{x^6 + 4}{x^6}$ using two methods:

1. Quotient Rule, and
2. Simplifying first.

Method 1: Using the Quotient Rule

The quotient rule states:

$\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}$

For $f(x) = \frac{x^6 + 4}{x^6}$:

• $u(x) = x^6 + 4$
• $v(x) = x^6$

Now we compute the derivatives:

• $u'(x) = 6x^5$ (since the derivative of $x^6$ is $6x^5$, and the derivative of a constant is 0)
• $v'(x) = 6x^5$

Now apply the quotient rule:

$f'(x) = \frac{x^6 \cdot 6x^5 - (x^6 + 4) \cdot 6x^5}{(x^6)^2}$

Simplify the numerator:

$f'(x) = \frac{6x^{11} - 6x^5(x^6 + 4)}{x^{12}} = \frac{6x^{11} - 6x^{11} - 24x^5}{x^{12}}$

This simplifies to:

$f'(x) = \frac{-24x^5}{x^{12}} = -\frac{24}{x^7}$

So, the derivative using the quotient rule is:

$f'(x) = -\frac{24}{x^7}$

The correct answer from the multiple choices provided would be option B, as it matches the quotient rule expression before simplifying.

Method 2: Simplifying First

We can rewrite $f(x) = \frac{x^6 + 4}{x^6}$ by splitting it into two terms:

$f(x) = \frac{x^6}{x^6} + \frac{4}{x^6} = 1 + 4x^{-6}$

Now, differentiate term by term:

• The derivative of $1$ is 0.
• The derivative of $4x^{-6}$ is $-24x^{-7}$.

Thus:

$f'(x) = -\frac{24}{x^7}$

So, both methods give the same result:

$f'(x) = -\frac{24}{x^7}$

Summary:

• Using the quotient rule, we find that $f'(x) = -\frac{24}{x^7}$.
• Simplifying first gives the same result: $f'(x) = -\frac{24}{x^7}$.
• The correct answer in the multiple-choice section is option B.

Would you like further details or clarifications on any steps?

Here are 5 related questions:

1. How do you apply the product rule for differentiation?
2. What is the chain rule in calculus, and when do you use it?
3. How can you differentiate a function with multiple variables?
4. What happens when you apply implicit differentiation to an equation?
5. How would you integrate a function like $x^{-7}$?

Tip: Always try simplifying expressions before applying differentiation rules like the quotient rule to make calculations easier!