{x^5}{(x^2+4)^2} find this
Different fractions

To find the derivative of the expression:

\[
y = \frac{x^5}{(x^2 + 4)^2}
\]

we'll use the **quotient rule** along with the **product rule** and **chain rule** as needed.

The **quotient rule** for differentiation states that:

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

Here:

- \( f(x) = x^5 \) and
- \( g(x) = (x^2 + 4)^2 \)

### Step 1: Differentiate \( f(x) = x^5 \)
\[
f'(x) = 5x^4
\]

### Step 2: Differentiate \( g(x) = (x^2 + 4)^2 \) using the chain rule
Let \( u(x) = x^2 + 4 \), so \( g(x) = u^2 \).

Now, differentiate \( u(x) = x^2 + 4 \):
\[
u'(x) = 2x
\]

Now apply the chain rule to differentiate \( g(x) = u^2 \):
\[
g'(x) = 2u \cdot u'(x) = 2(x^2 + 4) \cdot 2x = 4x(x^2 + 4)
\]

### Step 3: Apply the quotient rule
Now, substitute into the quotient rule formula:
\[
\frac{dy}{dx} = \frac{(5x^4)(x^2 + 4)^2 - (x^5)(4x(x^2 + 4))}{(x^2 + 4)^4}
\]

### Step 4: Simplify the expression
Simplify the numerator:
\[
\text{Numerator} = 5x^4(x^2 + 4)^2 - 4x^6(x^2 + 4)
\]

We can factor out \( x^4(x^2 + 4) \) from both terms in the numerator:
\[
\text{Numerator} = x^4(x^2 + 4) \left[ 5(x^2 + 4) - 4x^2 \right]
\]

Simplify the terms inside the brackets:
\[
5(x^2 + 4) - 4x^2 = 5x^2 + 20 - 4x^2 = x^2 + 20
\]

So, the numerator becomes:
\[
\text{Numerator} = x^4(x^2 + 4)(x^2 + 20)
\]

Now, the derivative is:
\[
\frac{dy}{dx} = \frac{x^4(x^2 + 4)(x^2 + 20)}{(x^2 + 4)^4}
\]

Finally, cancel one factor of \( (x^2 + 4) \) from the numerator and the denominator:
\[
\frac{dy}{dx} = \frac{x^4(x^2 + 20)}{(x^2 + 4)^3}
\]

This is the simplified derivative.

Would you like more details, or have any questions?

### Related Questions:
1. How do you apply the product rule in differentiation?
2. Can you explain the chain rule in more detail?
3. How does implicit differentiation work in calculus?
4. What are some common mistakes when using the quotient rule?
5. How does the power rule for differentiation work?

### Tip:
When differentiating complex expressions, always simplify as much as possible before applying rules like the quotient rule—it helps avoid unnecessary complications.

Learn how to find the derivative of x^5 / (x^2 + 4)^2 using the quotient rule, product rule, and chain rule. This step-by-step solution simplifies the complex differentiation process for undergraduate calculus students.

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