To find the derivative of the function $f(x) = \frac{x}{x^4 + 4}$ using the Quotient Rule, follow these steps:

The Quotient Rule:

The derivative of a function $f(x) = \frac{g(x)}{h(x)}$ is given by:

$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{h(x)^2}$

Where:

• $g(x) = x$
• $h(x) = x^4 + 4$

Step-by-step solution:

1. Find $g'(x)$:
   $g(x) = x$, so $g'(x) = 1$.

2. Find $h'(x)$:
   $h(x) = x^4 + 4$, so $h'(x) = 4x^3$.

3. Apply the Quotient Rule: $f'(x) = \frac{(1)(x^4 + 4) - (x)(4x^3)}{(x^4 + 4)^2}$

4. Simplify the numerator: $f'(x) = \frac{x^4 + 4 - 4x^4}{(x^4 + 4)^2}$ $f'(x) = \frac{-3x^4 + 4}{(x^4 + 4)^2}$

Thus, the derivative of the function is: $f'(x) = \frac{-3x^4 + 4}{(x^4 + 4)^2}$

Do you want more details or have any questions?

Here are 5 related questions:

1. What is the chain rule and how does it compare to the quotient rule?
2. How can you use the product rule to find derivatives?
3. What are some typical applications of the quotient rule in physics?
4. Can the quotient rule be used to find higher-order derivatives of this function?
5. How would the quotient rule change if the numerator was a more complex function?

Tip: Always simplify both the numerator and denominator after applying the quotient rule to avoid errors in further calculations.