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When you're dealing with a function that's a fraction, like \(f(x)=\frac{x^{3}-2}{x^{2}+1}\), you need the Quotient Rule to find its derivative. This rule helps us differentiate a function that is one function divided by another. The formula for the Quotient Rule is:\[\left(\frac{u}{v}\right)' = \frac{vu' - uv'}{v^2}\]Here, \(u\) is the numerator and \(v\) is the denominator of the fraction. Once you identify \(u\) and \(v\), differentiate both to get \(u'\) and \(v'\). Then, plug all these into the formula above to find the derivative. It’s a systematic method that helps make complex fractions more manageable.
Make sure to correctly identify and apply it step by step, ensuring no part of the function is forgotten.

The Power Rule is often your best friend when differentiating functions, especially polynomials. It's simple and incredibly useful. The rule states that if you have a function \(x^n\), its derivative is \(nx^{n-1}\). This means we multiply the exponent by the coefficient and subtract one from the exponent:- For \(u(x) = x^3 - 2\), differentiating gives \(u'(x) = 3x^2\) since \(3\cdot x^{3-1} = 3x^2\).- For \(v(x) = x^2 + 1\), it yields \(v'(x) = 2x\) because \(2\cdot x^{2-1} = 2x\).Apply the Power Rule to each part of the function separately.
It's a straightforward way to deal with each term individually before plugging them into the Quotient Rule or any other derivative rules.

After applying the Quotient Rule, you might end up with a complex fraction that needs simplifying. Simplifying expressions is crucial; it ensures that your solution is as neat and concise as possible.Start by distributing terms within the numerator:- For \(f'(x)=\frac{(x^{2}+1)3x^{2}-(x^{3}-2)2x}{(x^{2}+1)^{2}}\), distribute the coefficients, i.e., \(3x^2\) and \(2x\), across each term inside their respective parentheses.Next, combine like terms, which involves simplifying and reorganizing:- After distributing, you have \(3x^4 + 3x^2 - 2x^4 + 4x\).- Combine to get \(x^4 + 3x^2 + 4x\).These steps are straightforward but can greatly clarify the equation.
Simplification ensures that your derivative is correctly and clearly expressed, making it easier to work with in further calculations.