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

Ask by Moran Burgess. in the United States
Jan 24,2025

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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:** \[ h'(x) = \frac{2x^4 - 33x^2 - 15}{(x^2 - 5)^2} \]

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To use the quotient rule for differentiation, remember that it states \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \), where \( u \) and \( v \) are both functions of \( x \). So, if you have the expression \( \frac{f(x)}{g(x)} \), calculate the derivatives \( f'(x) \) and \( g'(x) \), then plug them into the formula for the quotient rule. Next, ensure that \( v \neq 0 \) in the original function, as division by zero is a no-go! Also, keep an eye out for simplification opportunities; often, after applying the quotient rule, you can factor or cancel terms to make your final derivative tidier.

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