$$ 1 \leftarrow $$ Divide the expression using the quotient rule.

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UpStudy AI · Accepted Answer

Sure, I'd be happy to help you apply the **quotient rule**! However, I'll need the specific expression you'd like to differentiate.

**The Quotient Rule** is used to differentiate expressions of the form:

$$
\frac{f(x)}{g(x)}
$$

The formula for the derivative is:

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

### **Example:**

Let's say you want to differentiate the following function using the quotient rule:

$$
h(x) = \frac{2x^3 + 3x}{x^2 - 5}
$$

**Step 1: Identify $$ f(x) $$ and $$ g(x) $$**
$$
f(x) = 2x^3 + 3x \
g(x) = x^2 - 5
$$

**Step 2: Find $$ f'(x) $$ and $$ g'(x) $$**
$$
f'(x) = 6x^2 + 3 \
g'(x) = 2x
$$

**Step 3: Apply the Quotient Rule**
$$
h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \
h'(x) = \frac{(6x^2 + 3)(x^2 - 5) - (2x^3 + 3x)(2x)}{(x^2 - 5)^2}
$$

**Step 4: Simplify the Expression**
$$
h'(x) = \frac{6x^4 - 30x^2 + 3x^2 - 15 - 4x^4 - 6x^2}{(x^2 - 5)^2} \
h'(x) = \frac{(6x^4 - 4x^4) + (-30x^2 + 3x^2 - 6x^2) - 15}{(x^2 - 5)^2} \
h'(x) = \frac{2x^4 - 33x^2 - 15}{(x^2 - 5)^2}
$$

**Final Answer:**
$$
h'(x) = \frac{2x^4 - 33x^2 - 15}{(x^2 - 5)^2}
$$

---

If you provide the specific expression you're working with, I can guide you through the steps tailored to that function!
To differentiate the expression $$ \frac{2x^3 + 3x}{x^2 - 5} $$ using the quotient rule, follow these steps: 1. **Identify $$ f(x) $$ and $$ g(x) $$:** $$ f(x) = 2x^3 + 3x \ g(x) = x^2 - 5 $$ 2. **Find $$ f'(x) $$ and $$ g'(x) $$:** $$ f'(x) = 6x^2 + 3 \ g'(x) = 2x $$ 3. **Apply the Quotient Rule:** $$ h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \ h'(x) = \frac{(6x^2 + 3)(x^2 - 5) - (2x^3 + 3x)(2x)}{(x^2 - 5)^2} $$ 4. **Simplify the Expression:** $$ h'(x) = \frac{2x^4 - 33x^2 - 15}{(x^2 - 5)^2} $$ **Final Answer:** $$ h'(x) = \frac{2x^4 - 33x^2 - 15}{(x^2 - 5)^2} $$

Divide the expression using the quotient rule.

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