Differentiate f(x) = 𝑥3 +1 4𝑥2+1 using the quotient rule

To differentiate \( f(x) = \frac{x^3 + 1}{4x^2 + 1} \) using the quotient rule, we apply the formula:

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

Where:
- \( u(x) = x^3 + 1 \)
- \( v(x) = 4x^2 + 1 \)

### Step 1: Differentiate the numerator \( u(x) \)
\[
u'(x) = \frac{d}{dx} (x^3 + 1) = 3x^2
\]

### Step 2: Differentiate the denominator \( v(x) \)
\[
v'(x) = \frac{d}{dx} (4x^2 + 1) = 8x
\]

### Step 3: Apply the quotient rule
Now plug everything into the quotient rule formula:

\[
f'(x) = \frac{(4x^2 + 1)(3x^2) - (x^3 + 1)(8x)}{(4x^2 + 1)^2}
\]

### Step 4: Simplify the numerator
First, expand both terms in the numerator:

\[
(4x^2 + 1)(3x^2) = 12x^4 + 3x^2
\]
\[
(x^3 + 1)(8x) = 8x^4 + 8x
\]

So the numerator becomes:
\[
(12x^4 + 3x^2) - (8x^4 + 8x) = 12x^4 + 3x^2 - 8x^4 - 8x = 4x^4 + 3x^2 - 8x
\]

### Step 5: Write the final result
The derivative of \( f(x) \) is:

\[
f'(x) = \frac{4x^4 + 3x^2 - 8x}{(4x^2 + 1)^2}
\]

Let me know if you'd like further details or clarification!

### Related questions:
1. How would the quotient rule change if \( f(x) \) had more complicated polynomials?
2. Can you show how to use the product rule in combination with the quotient rule?
3. What happens if the denominator has a higher degree than the numerator?
4. How would you compute the second derivative of \( f(x) \)?
5. What is the physical interpretation of the derivative of this function?

**Tip:** Always simplify the numerator carefully after applying the quotient rule, as this reduces the complexity of the derivative expression.

Differentiate f(x) = 𝑥^3 + 1 / 4𝑥^2 + 1 using the quotient rule

Learn how to differentiate the function f(x) = (x^3 + 1) / (4x^2 + 1) using the quotient rule. This problem covers key calculus concepts, including the application of the quotient rule and simplifying derivatives. Suitable for students in Grades 11-12 or early university-level calculus.

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