Master the Quotient Rule: A Step-by-Step Guide to Finding Derivatives of Quotient Functions

Quotient Rule

The quotient rule is a formula used to find the derivative of a quotient of two functions

The quotient rule is a formula used to find the derivative of a quotient of two functions. When you have a function that is in the form of f(x) = g(x) / h(x), where g(x) and h(x) are differentiable functions, the quotient rule allows you to find the derivative of f(x) with respect to x.

Mathematically, the quotient rule can be stated as follows:

If f(x) = g(x) / h(x), then the derivative of f(x) with respect to x, denoted as f'(x), is given by:

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

Let’s break down the equation and understand each part:

– g'(x) is the derivative of the numerator function g(x) with respect to x.
– h'(x) is the derivative of the denominator function h(x) with respect to x.
– (h(x))^2 represents the square of the denominator function h(x).

To apply the quotient rule, follow these steps:

1. Differentiate the numerator function g(x) with respect to x to find g'(x).
2. Differentiate the denominator function h(x) with respect to x to find h'(x).
3. Substitute the values of g'(x), g(x), h'(x), and h(x) into the quotient rule formula.
4. Simplify the expression to get the final derivative f'(x).

It’s important to note that the quotient rule is derived from the product rule and chain rule, which are other important rules in calculus. So having a good understanding of these rules is helpful when using the quotient rule.

Overall, the quotient rule is a powerful technique for finding the derivative of a quotient of functions, enabling you to solve various problems involving rates of change, optimization, and curve sketching.

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