The Quotient Rule: How to Find the Derivative of a Function Using Calculus

quotient rule

The quotient rule is a rule used in calculus to find the derivative of a function that is the quotient of two other functions

The quotient rule is a rule used in calculus to find the derivative of a function that is the quotient of two other functions. It is specifically used when we need to differentiate a function that can be written as f(x) = g(x) / h(x).

To apply the quotient rule, follow these steps:

1. Identify the numerator function, g(x), and the denominator function, h(x), in your overall function f(x) = g(x) / h(x).

2. Differentiate the numerator function, g(x), to find g'(x), which represents the derivative of g(x) with respect to x.

3. Differentiate the denominator function, h(x), to find h'(x), which represents the derivative of h(x) with respect to x.

4. Use the quotient rule formula to find f'(x), which represents the derivative of f(x) with respect to x. The formula is:

f'(x) = (h(x) * g'(x) – g(x) * h'(x)) / (h(x))^2

5. Simplify the resulting expression, if possible, to obtain the final derivative.

To understand the concept better, let’s work through an example:

Example:
Find the derivative of f(x) = (3x^2 – 1) / (2x)

Solution:
1. Identify the numerator function g(x) = 3x^2 – 1 and the denominator function h(x) = 2x.

2. Differentiate the numerator function g(x):
g'(x) = 6x.

3. Differentiate the denominator function h(x):
h'(x) = 2.

4. Apply the quotient rule formula:
f'(x) = (h(x) * g'(x) – g(x) * h'(x)) / (h(x))^2
= (2x * 6x – (3x^2 – 1) * 2) / (2x)^2
= (12x^2 – 2(3x^2 – 1)) / (4x^2)
= (12x^2 – 6x^2 + 2) / (4x^2)
= (6x^2 + 2) / (4x^2)
= (3x^2 + 1) / (2x^2).

5. The final derivative is (3x^2 + 1) / (2x^2).

That’s it! You have successfully applied the quotient rule to find the derivative of the given function.

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