How do you differentiate a function divided by a function?

To differentiate a function divided by another function, you use the quotient rule.

Here's how the quotient rule works:

If you have a function h(x) defined as:

h(x) = f(x) / g(x)

Then the derivative of h(x), denoted as h'(x), is given by:

h'(x) = [g(x) f'(x) - f(x) g'(x)] / [g(x)]²

Where:

• f'(x) is the derivative of f(x)
• g'(x) is the derivative of g(x)

Explanation and Breakdown:

The quotient rule is a fundamental rule in calculus used to find the derivative of a function that is the ratio of two other functions. It's essential for differentiating expressions where one function is divided by another.

Here's a step-by-step breakdown:

1. Identify f(x) and g(x): Determine which function is in the numerator (f(x)) and which is in the denominator (g(x)).

2. Find the derivatives f'(x) and g'(x): Calculate the derivatives of both the numerator and denominator functions. You may need to apply other differentiation rules (power rule, chain rule, etc.) to find these derivatives.

3. Apply the formula: Substitute f(x), g(x), f'(x), and g'(x) into the quotient rule formula:

   h'(x) = [g(x) f'(x) - f(x) g'(x)] / [g(x)]²

4. Simplify: Simplify the resulting expression algebraically to obtain the final derivative h'(x).

Example:

Let's say h(x) = (x² + 1) / (x - 2)

1. f(x) = x² + 1
2. g(x) = x - 2

Now, find the derivatives:

1. f'(x) = 2x
2. g'(x) = 1

Apply the quotient rule:

h'(x) = [(x - 2) (2x) - (x² + 1) (1)] / (x - 2)²

Simplify:

h'(x) = [2x² - 4x - x² - 1] / (x - 2)²

h'(x) = (x² - 4x - 1) / (x - 2)²

Therefore, the derivative of h(x) = (x² + 1) / (x - 2) is h'(x) = (x² - 4x - 1) / (x - 2)².

Key Takeaways:

• The quotient rule is crucial when dealing with functions that are the ratio of two other functions.
• Remember the order of operations in the numerator: (g(x) f'(x)) - (f(x) g'(x)). Switching the order will result in the wrong answer.
• Always simplify the final expression as much as possible.