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Quotient Rule: How to Find Derivatives of Fractions

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The Quotient Rule: A Guide to Derivatives of Fractions

In the exciting world of calculus, the quotient rule is a fundamental tool for finding the derivatives of functions that are expressed as fractions. It’s like having a secret weapon to tackle complex mathematical problems.

What is the Quotient Rule?

The quotient rule states that the derivative of a quotient (a fraction where the numerator and denominator are both functions of x) is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

Let’s break it down with a formula:

**d/dx [f(x)/g(x)] = [g(x) * f'(x) – f(x) * g'(x)] / [g(x)]²**

Where:

  • f(x) is the numerator of the fraction
  • g(x) is the denominator of the fraction
  • f'(x) is the derivative of the numerator
  • g'(x) is the derivative of the denominator

Step-by-Step Guide

To use the quotient rule, follow these steps:

  1. **Identify f(x) and g(x):** Determine the numerator and denominator of your fraction.
  2. **Find f'(x) and g'(x):** Calculate the derivatives of both the numerator and denominator.
  3. **Apply the formula:** Substitute the values of f(x), g(x), f'(x), and g'(x) into the quotient rule formula.
  4. **Simplify:** Simplify the resulting expression to obtain the derivative of the original fraction.

Example

Let’s find the derivative of the following function using the quotient rule:

**f(x) = (x² + 1) / (x – 2)**

  1. **f(x) = x² + 1** (numerator)
  2. **g(x) = x – 2** (denominator)
  3. **f'(x) = 2x** (derivative of the numerator)
  4. **g'(x) = 1** (derivative of the denominator)

Now, let’s plug these values into the quotient rule formula:

**d/dx [(x² + 1) / (x – 2)] = [(x – 2) * 2x – (x² + 1) * 1] / (x – 2)²**

Simplifying the expression, we get:

**d/dx [(x² + 1) / (x – 2)] = (2x² – 4x – x² – 1) / (x – 2)²**

**d/dx [(x² + 1) / (x – 2)] = (x² – 4x – 1) / (x – 2)²**

Therefore, the derivative of the function (x² + 1) / (x – 2) is (x² – 4x – 1) / (x – 2)².

Conclusion

The quotient rule is a powerful tool for finding derivatives of fractions in calculus. By following the steps outlined above, you can confidently apply this rule to solve a wide range of problems. Remember to practice and master the quotient rule to excel in your calculus journey.

Higher Education