Given the function y, equals, start fraction, x, to the power 4 , plus, 2, divided by, 4, minus, 3, x, to the power 4 , end fraction, commay= 
4−3x 
4
 
x 
4
 +2
​
 , find start fraction, d, y, divided by, d, x, end fraction 
dx
dy
​
  in simplified form.

We are tasked with finding the derivative \( \frac{dy}{dx} \) of the function:

\[
y = \frac{x^4 + 2}{4 - 3x^4}
\]

This is a quotient, so we will apply the **quotient rule**, which states:

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

Where:

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

### Step 1: Find \( u'(x) \) and \( v'(x) \)

- \( u(x) = x^4 + 2 \), so \( u'(x) = 4x^3 \).
- \( v(x) = 4 - 3x^4 \), so \( v'(x) = -12x^3 \).

### Step 2: Apply the quotient rule

Now, using the quotient rule:

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

### Step 3: Simplify the numerator

First, expand the terms in the numerator:

\[
(4 - 3x^4)(4x^3) = 16x^3 - 12x^7
\]

\[
(x^4 + 2)(-12x^3) = -12x^7 - 24x^3
\]

Now, substitute these into the numerator:

\[
(16x^3 - 12x^7) - (-12x^7 - 24x^3) = 16x^3 - 12x^7 + 12x^7 + 24x^3
\]

Simplifying:

\[
16x^3 + 24x^3 = 40x^3
\]

So the numerator is:

\[
40x^3
\]

### Step 4: Final expression

Thus, the derivative becomes:

\[
\frac{dy}{dx} = \frac{40x^3}{(4 - 3x^4)^2}
\]

This is the simplified form of \( \frac{dy}{dx} \).

Would you like to see further details on any step?

Here are some related questions:
1. How would you apply the product rule in calculus?
2. Can you explain when to use the chain rule versus the quotient rule?
3. How does the quotient rule differ from the product rule?
4. What is the general form of the derivative of a rational function?
5. How do higher-order derivatives of this function behave?

**Tip**: Always identify the function's form (sum, product, quotient) before deciding which differentiation rule to apply!

Given the function y = (x^4 + 2) / (4 - 3x^4), find dy/dx in simplified form.

Find the derivative of the function y = (x^4 + 2) / (4 - 3x^4) using the quotient rule. This step-by-step solution demonstrates how to apply calculus concepts such as the quotient rule and power rule. Ideal for students in Grades 11-12 studying advanced calculus.

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