Make a simple qoutient rule example with solution for calculus

Here’s a simple example of applying the **Quotient Rule** in calculus:

### Problem:
Find the derivative of the function:
\[
f(x) = \frac{3x^2 + 4}{x}
\]

### Solution:
To apply the **Quotient Rule**, which states:
\[
\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{[v(x)]^2}
\]
we need to identify \( u(x) \) (the numerator) and \( v(x) \) (the denominator).

In this case:
- \( u(x) = 3x^2 + 4 \)
- \( v(x) = x \)

Now, differentiate both \( u(x) \) and \( v(x) \):
- \( u'(x) = \frac{d}{dx}(3x^2 + 4) = 6x \)
- \( v'(x) = \frac{d}{dx}(x) = 1 \)

Now, apply the quotient rule:
\[
f'(x) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{[v(x)]^2}
\]
Substitute \( u(x), v(x), u'(x), v'(x) \):
\[
f'(x) = \frac{x(6x) - (3x^2 + 4)(1)}{x^2}
\]
Simplify the numerator:
\[
f'(x) = \frac{6x^2 - (3x^2 + 4)}{x^2} = \frac{6x^2 - 3x^2 - 4}{x^2} = \frac{3x^2 - 4}{x^2}
\]

Now, simplify the entire expression:
\[
f'(x) = 3 - \frac{4}{x^2}
\]

### Final Answer:
\[
f'(x) = 3 - \frac{4}{x^2}
\]

Would you like more details or have any other questions?

### Related Questions:
1. How do you apply the product rule in calculus?
2. Can you explain when to use the chain rule vs. the quotient rule?
3. What happens when the denominator has a more complex function?
4. Can you show a more complex quotient rule example?
5. How do the quotient rule and the chain rule work together?

### Tip:
Always simplify your derivative expression fully after applying the quotient rule to make it easier to interpret the result.

Make a simple quotient rule example with solution for calculus

Explore a simple example of the quotient rule in calculus with a detailed step-by-step solution. Learn how to find the derivative of the function f(x) = (3x^2 + 4) / x, using the quotient rule. Perfect for students in Grades 11-12 or early college studying calculus.

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