\( 1 \leftarrow \quad \) Divide the expression using the quotient rule.

Certainly! The **quotient rule** is a fundamental technique in calculus used to find the derivative of a function that is the ratio of two differentiable functions. ### Quotient Rule Formula If you have a function \( f(x) \) defined as: \[ f(x) = \frac{g(x)}{h(x)} \] where both \( g(x) \) and \( h(x) \) are differentiable functions, then the derivative of \( f(x) \), denoted as \( f'(x) \), is given by: \[ f'(x) = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{[h(x)]^2} \] ### Step-by-Step Application Let's walk through the steps to apply the quotient rule with an example. **Example:** Differentiate the function \[ f(x) = \frac{2x^3 + 3x}{x^2 - 5} \] **Step 1: Identify \( g(x) \) and \( h(x) \)** \[ g(x) = 2x^3 + 3x \\ h(x) = x^2 - 5 \] **Step 2: Find the derivatives \( g'(x) \) and \( h'(x) \)** \[ g'(x) = \frac{d}{dx}(2x^3 + 3x) = 6x^2 + 3 \\ h'(x) = \frac{d}{dx}(x^2 - 5) = 2x \] **Step 3: Apply the Quotient Rule Formula** \[ f'(x) = \frac{(6x^2 + 3)(x^2 - 5) - (2x^3 + 3x)(2x)}{(x^2 - 5)^2} \] **Step 4: Simplify the Expression** First, expand the numerator: \[ (6x^2 + 3)(x^2 - 5) = 6x^4 - 30x^2 + 3x^2 - 15 = 6x^4 - 27x^2 - 15 \] \[ (2x^3 + 3x)(2x) = 4x^4 + 6x^2 \] Subtract the second product from the first: \[ 6x^4 - 27x^2 - 15 - (4x^4 + 6x^2) = 2x^4 - 33x^2 - 15 \] So, the derivative is: \[ f'(x) = \frac{2x^4 - 33x^2 - 15}{(x^2 - 5)^2} \] ### Summary To **divide** an expression using the **quotient rule**: 1. **Identify** the numerator function \( g(x) \) and the denominator function \( h(x) \). 2. **Differentiate** both \( g(x) \) and \( h(x) \) to find \( g'(x) \) and \( h'(x) \). 3. **Apply** the quotient rule formula: \[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} \] 4. **Simplify** the resulting expression as needed. If you have a specific expression you'd like to work through, feel free to share it, and I can guide you through the process!