Finding the derivative of a fraction, also known as a quotient, is a fundamental skill in calculus. It involves applying the quotient rule, a specific formula designed for differentiating expressions where one function is divided by another. This article will guide you through the process of finding the derivative of a fraction, explaining the quotient rule in detail and providing step-by-step examples.
Understanding the Quotient Rule
The quotient rule states that the derivative of a fraction, where the numerator is u(x) and the denominator is v(x), is given by:
d/dx [u(x) / v(x)] = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2
This rule essentially involves taking the derivative of the numerator, multiplying it by the denominator, subtracting the derivative of the denominator multiplied by the numerator, and then dividing the entire result by the square of the denominator.
Step-by-Step Guide to Finding the Derivative of a Fraction
To find the derivative of a fraction, follow these steps:
- Identify the numerator and denominator: Clearly identify the functions representing the numerator, u(x), and the denominator, v(x).
- Find the derivatives of the numerator and denominator: Calculate the derivative of the numerator, u'(x), and the derivative of the denominator, v'(x).
- Apply the quotient rule: Substitute the identified functions and their derivatives into the quotient rule formula.
- Simplify the expression: Simplify the resulting expression by combining like terms and performing any necessary algebraic manipulations.
Examples:
Example 1:
Find the derivative of (x^2 + 3x) / (2x - 1)
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Identify the numerator and denominator:
- u(x) = x^2 + 3x
- v(x) = 2x - 1
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Find the derivatives:
- u'(x) = 2x + 3
- v'(x) = 2
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Apply the quotient rule:
- d/dx [(x^2 + 3x) / (2x - 1)] = [(2x - 1)(2x + 3) - (x^2 + 3x)(2)] / [(2x - 1)^2]
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Simplify the expression:
- = (4x^2 + 4x - 3 - 2x^2 - 6x) / (4x^2 - 4x + 1)
- = (2x^2 - 2x - 3) / (4x^2 - 4x + 1)
Therefore, the derivative of (x^2 + 3x) / (2x - 1) is (2x^2 - 2x - 3) / (4x^2 - 4x + 1).
Example 2:
Find the derivative of (sin(x)) / (x^3 + 1)
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Identify the numerator and denominator:
- u(x) = sin(x)
- v(x) = x^3 + 1
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Find the derivatives:
- u'(x) = cos(x)
- v'(x) = 3x^2
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Apply the quotient rule:
- d/dx [(sin(x)) / (x^3 + 1)] = [(x^3 + 1)(cos(x)) - (sin(x))(3x^2)] / [(x^3 + 1)^2]
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Simplify the expression:
- = (x^3cos(x) + cos(x) - 3x^2sin(x)) / (x^6 + 2x^3 + 1)
Therefore, the derivative of (sin(x)) / (x^3 + 1) is (x^3cos(x) + cos(x) - 3x^2sin(x)) / (x^6 + 2x^3 + 1).
Common Mistakes to Avoid
When finding the derivative of a fraction, it is crucial to avoid these common mistakes:
- Incorrectly applying the quotient rule: Ensure you correctly identify the numerator and denominator, their respective derivatives, and substitute them accurately into the formula.
- Neglecting to square the denominator: Remember to square the denominator (v(x)) after substituting the values into the quotient rule formula.
- Simplifying the expression incorrectly: Carefully perform any algebraic manipulations to simplify the derivative to its simplest form.
Conclusion
Finding the derivative of a fraction requires applying the quotient rule, a fundamental concept in calculus. By understanding and applying the steps outlined in this article, you can successfully differentiate any fraction and navigate this crucial aspect of calculus. Remember to identify the numerator and denominator, calculate their derivatives, apply the quotient rule formula, and simplify the resulting expression to find the derivative of the fraction.
