The Quotient Rule | How to Find the Derivative of a Function Quotient

d/dx[f(x)/g(x)]

To find the derivative of a function that is a quotient of two functions, such as f(x)/g(x), we can use the quotient rule

To find the derivative of a function that is a quotient of two functions, such as f(x)/g(x), we can use the quotient rule. The quotient rule states that the derivative of a quotient is given by the following formula:

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

Let’s break down each part of the formula:

– f(x) is the numerator of the function.
– g(x) is the denominator of the function.

Now, let’s differentiate the numerator and denominator separately:

– d/dx[f(x)] represents the derivative of the numerator, f(x).
– d/dx[g(x)] represents the derivative of the denominator, g(x).

Once we have these derivatives, we can substitute them back into the quotient rule formula to find the derivative of f(x)/g(x).

Here’s an example to illustrate how to find the derivative:

Suppose we have the function y = (2x^2 + 3x + 1)/(x^3 – 5x).

Let’s differentiate the numerator and denominator separately:

– d/dx[2x^2 + 3x + 1] = 4x + 3
– d/dx[x^3 – 5x] = 3x^2 – 5

Now, substitute these values into the quotient rule formula:

d/dx[(2x^2 + 3x + 1)/(x^3 – 5x)] = [(x^3 – 5x)*(4x + 3) – (2x^2 + 3x + 1)*(3x^2 – 5)] / [(x^3 – 5x)^2]

After simplifying this expression, you will have the derivative of the given function.

Remember that the quotient rule is a useful tool when finding the derivative of functions that are expressed as a quotient of two functions.

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