Step 1: Given Function

We are given the function: $f(x) = \frac{2 - x^2}{4 + x^2}$

Step 2: Quotient Rule Formula

The Quotient Rule is used to differentiate a function of the form: $f(x) = \frac{g(x)}{h(x)}$ The derivative $f'(x)$ is given by: $f'(x) = \frac{h(x)g'(x) - g(x)h'(x)}{[h(x)]^2}$ Where:

• $g(x) = 2 - x^2$
• $h(x) = 4 + x^2$

Step 3: Find $g'(x)$ and $h'(x)$

First, find the derivatives of $g(x)$ and $h(x)$:

• $g(x) = 2 - x^2$, so $g'(x) = -2x$
• $h(x) = 4 + x^2$, so $h'(x) = 2x$

Step 4: Apply the Quotient Rule

Now apply the quotient rule: $f'(x) = \frac{(4 + x^2)(-2x) - (2 - x^2)(2x)}{(4 + x^2)^2}$ Simplify the numerator: $f'(x) = \frac{(-2x)(4 + x^2) - (2 - x^2)(2x)}{(4 + x^2)^2}$ $f'(x) = \frac{-8x - 2x^3 - 4x + 2x^3}{(4 + x^2)^2}$ Combine like terms: $f'(x) = \frac{-12x}{(4 + x^2)^2}$

Step 5: Find $f'(1)$

Now substitute $x = 1$ into the expression for $f'(x)$: $f'(1) = \frac{-12(1)}{(4 + 1^2)^2}$ $f'(1) = \frac{-12}{(4 + 1)^2} = \frac{-12}{5^2} = \frac{-12}{25}$

So, the derivative at $x = 1$ is: $f'(1) = \frac{-12}{25}$

Final Answer:

• $f'(x) = \frac{-12x}{(4 + x^2)^2}$
• $f'(1) = \frac{-12}{25}$

Would you like more details on any of the steps?

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5 Related Questions:

1. What is the chain rule and how does it differ from the quotient rule?
2. Can the quotient rule be applied to functions with more complex numerators and denominators?
3. How does the product rule relate to the quotient rule in calculus?
4. How do higher-order derivatives (second derivative, third derivative) apply when using the quotient rule?
5. What are some common mistakes to avoid when applying the quotient rule?

Tip:

Always simplify both the numerator and denominator separately when using the quotient rule to avoid algebraic mistakes.