quotient rule
The quotient rule is a formula used in calculus to find the derivative of a function that is the ratio of two other functions
The quotient rule is a formula used in calculus to find the derivative of a function that is the ratio of two other functions. It is particularly useful when finding the derivative of a function that cannot be simplified by factoring out a common factor.
Let’s say we have two functions, f(x) and g(x), and we want to find the derivative of their ratio, h(x) = f(x) / g(x). The quotient rule states that the derivative of h(x) is given by the following formula:
h'(x) = (g(x) * f'(x) – f(x) * g'(x)) / g(x)^2
To apply the quotient rule, we need to know the derivatives of both f(x) and g(x), which are denoted as f'(x) and g'(x) respectively.
Let’s go through an example to see how the quotient rule is used:
Example:
Consider the function h(x) = (3x^2 + 2x + 1) / (2x + 1)
Step 1: Identify f(x) and g(x)
In this case, f(x) = 3x^2 + 2x + 1 and g(x) = 2x + 1
Step 2: Find f'(x) and g'(x)
To find f'(x), we take the derivative of f(x) term by term. In this case:
f'(x) = d/dx (3x^2) + d/dx (2x) + d/dx (1)
= 6x + 2 + 0
= 6x + 2
To find g'(x), we take the derivative of g(x):
g'(x) = d/dx (2x) + d/dx (1)
= 2 + 0
= 2
Step 3: Apply the quotient rule
Using the formula h'(x) = (g(x) * f'(x) – f(x) * g'(x)) / g(x)^2, we have:
h'(x) = ((2x + 1) * (6x + 2) – (3x^2 + 2x + 1) * 2) / (2x + 1)^2
= (12x^2 + 4x + 6x + 2 – 6x^2 – 4x – 2) / (2x + 1)^2
= (6x^2 + 4x) / (2x + 1)^2
So, the derivative of h(x) is (6x^2 + 4x) / (2x + 1)^2.
That’s how the quotient rule is used to find the derivative of a function that is the ratio of two other functions. It is an important tool in calculus for solving problems involving rates of change and optimization.
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