How to use the quotient rule?
Understand the Problem
The question is asking for guidance on how to apply the quotient rule, which is a method for differentiating a function that is the quotient of two other functions. The quotient rule states that if you have a function that is defined as the ratio of two functions, then the derivative can be calculated using a specific formula.
Answer
The derivative using the quotient rule is given by $y' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$.
Answer for screen readers
The derivative of the function $y = \frac{f(x)}{g(x)}$ using the quotient rule is: $$ y' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} $$
Steps to Solve
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Identify the functions Identify the numerator and denominator of the quotient. If we have a function $y = \frac{f(x)}{g(x)}$, then $f(x)$ is the numerator and $g(x)$ is the denominator.
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Apply the quotient rule The quotient rule states that the derivative of a quotient of two functions $y = \frac{f(x)}{g(x)}$ can be calculated using the formula: $$ y' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} $$
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Calculate derivatives of the functions Find the derivatives $f'(x)$ and $g'(x)$ separately. This requires using basic differentiation rules.
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Substitute back into the quotient rule formula Insert the derivatives and original functions into the quotient rule formula. This will give you an expression for $y'$.
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Simplify the expression if necessary If possible, simplify the expression to make it cleaner or easier to understand.
The derivative of the function $y = \frac{f(x)}{g(x)}$ using the quotient rule is: $$ y' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} $$
More Information
The quotient rule is a helpful tool in calculus for finding the derivative of a function constructed as a ratio. It emphasizes the need to apply the product of the derivative of the numerator and the original denominator minus the product of the original numerator and the derivative of the denominator, all divided by the square of the denominator.
Tips
- Forgetting to square the denominator in the final formula.
- Mixing up $f(x)$ and $g(x)$ when substituting back into the formula.
- Not simplifying the answer which can lead to a more complicated expression.
