Quotient Rule
The quotient rule is a rule used in calculus to differentiate the quotient of two functions
The quotient rule is a rule used in calculus to differentiate the quotient of two functions. It helps us find the derivative of a function that is expressed as the ratio of two other functions.
If we have a function f(x) = g(x) / h(x), where both g(x) and h(x) are differentiable functions, the quotient rule allows us to find the derivative of f(x).
The quotient rule states that the derivative of f(x) is given by the formula:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2
In simpler terms, the derivative of the quotient of two functions is found by taking the derivative of the numerator multiplied by the denominator, minus the original numerator multiplied by the derivative of the denominator, all divided by the square of the denominator.
Let’s go through an example to illustrate the use of the quotient rule:
Example:
Consider the function f(x) = x^2 / (3x – 2).
To find the derivative of f(x), we need to use the quotient rule.
First, we find the derivative of the numerator, which is g'(x) = 2x.
Next, we find the derivative of the denominator, which is h'(x) = 3.
Now, using the quotient rule formula, we can find the derivative of f(x):
f'(x) = (2x * (3x – 2) – x^2 * 3) / ((3x – 2)^2)
Simplifying this expression further gives us:
f'(x) = (6x^2 – 4x – 3x^2) / (9x^2 – 12x + 4)
Combining like terms, we have:
f'(x) = (3x^2 – 4x) / (9x^2 – 12x + 4)
So, the derivative of f(x) is (3x^2 – 4x) / (9x^2 – 12x + 4).
This is how we use the quotient rule to find the derivative of a function expressed as the ratio of two other functions.
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