The Quotient Rule: A Deeper Dive with a Complex Example

In the realm of calculus, derivatives are powerful tools for understanding the rate of change of functions. While the basic rules of differentiation are relatively straightforward, some functions require more sophisticated techniques. One such technique is the quotient rule, which is essential for finding the derivatives of functions that involve division.

The quotient rule states that the derivative of a quotient of two functions, f(x) and g(x), is:

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

Let’s break down this formula:

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

While the formula might look intimidating at first, applying it in practice is quite manageable. Let’s illustrate with a complex example.

Example: Finding the Derivative of a Complex Quotient

Let’s find the derivative of the following function:

f(x) = (x² + 3x) / (x³ – 2)

We can apply the quotient rule to solve this problem.

1. Identify f(x) and g(x):
• f(x) = x² + 3x
• g(x) = x³ – 2
2. Find f'(x) and g'(x):
• f'(x) = 2x + 3
• g'(x) = 3x²
3. Substitute the values into the quotient rule formula:

d/dx [f(x) / g(x)] = [(x³ – 2) * (2x + 3) – (x² + 3x) * (3x²)] / [(x³ – 2)²]

4. Simplify the expression:

d/dx [f(x) / g(x)] = (2x⁴ + 3x³ – 4x – 6 – 3x⁴ – 9x³) / (x⁶ – 4x³ + 4)

d/dx [f(x) / g(x)] = (-x⁴ – 6x³ – 4x – 6) / (x⁶ – 4x³ + 4)

Therefore, the derivative of the function f(x) = (x² + 3x) / (x³ – 2) is (-x⁴ – 6x³ – 4x – 6) / (x⁶ – 4x³ + 4).

While this example demonstrates the application of the quotient rule, it’s important to remember that practice is key to mastering this concept. By working through different examples and understanding the logic behind the formula, you can confidently apply the quotient rule to solve complex derivative problems.

Additional Tips for Applying the Quotient Rule

• Always identify f(x) and g(x) clearly. This helps avoid confusion when substituting values into the formula.
• Be careful with signs. The quotient rule involves subtraction, so ensure you’re subtracting the correct terms.
• Simplify your answer. After applying the formula, it’s often possible to simplify the resulting expression.
• Practice, practice, practice! The more you apply the quotient rule, the more comfortable you’ll become with it.

The quotient rule is a valuable tool in calculus, and with a little practice, you can master its application to solve a wide range of derivative problems.