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.