Find f′(x) using the **QUOTIENT RULE** iff(x)=2−x24+x2.f′(x)= 

Find f′(1).

f′(1)=

We are given the function:

\[
f(x) = \frac{2 - x^2}{4 + x^2}
\]

To find \( f'(x) \) using the **Quotient Rule**, we use the formula:

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

where:
- \( u(x) = 2 - x^2 \)
- \( v(x) = 4 + x^2 \)

### Step 1: Differentiate \( u(x) \) and \( v(x) \)

- \( u'(x) = \frac{d}{dx}(2 - x^2) = -2x \)
- \( v'(x) = \frac{d}{dx}(4 + x^2) = 2x \)

### Step 2: Apply the Quotient Rule

\[
f'(x) = \frac{(4 + x^2)(-2x) - (2 - x^2)(2x)}{(4 + x^2)^2}
\]

Simplify the numerator:

\[
= \frac{-(4 + x^2)2x - (2 - x^2)2x}{(4 + x^2)^2}
\]
\[
= \frac{-8x - 2x^3 - 4x + 2x^3}{(4 + x^2)^2}
\]
\[
= \frac{-12x}{(4 + x^2)^2}
\]

Thus, the derivative is:

\[
f'(x) = \frac{-12x}{(4 + x^2)^2}
\]

### Step 3: Find \( f'(1) \)

Substitute \( x = 1 \) into \( f'(x) \):

\[
f'(1) = \frac{-12(1)}{(4 + 1^2)^2} = \frac{-12}{(4 + 1)^2} = \frac{-12}{5^2} = \frac{-12}{25}
\]

Thus,

\[
f'(1) = \frac{-12}{25}
\]

Would you like further details or have any questions?

### 5 Related Questions:
1. How do we apply the product rule and quotient rule together in complex problems?
2. What are some examples where the quotient rule is more efficient than simplifying before differentiating?
3. What is the product rule, and how is it different from the quotient rule?
4. Can the quotient rule be applied to higher-order derivatives?
5. How does the quotient rule work for functions involving trigonometric terms?

### Tip:
When applying the quotient rule, carefully keep track of signs when simplifying the numerator to avoid common mistakes.

Find f′(x) using the QUOTIENT RULE if f(x)= (2 − x^2)/(4 + x^2). Find f′(1).

Learn how to find the derivative of the function f(x) = (2 - x^2)/(4 + x^2) using the Quotient Rule. This step-by-step guide simplifies the process, including the formula and calculation of f′(1). Ideal for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation