quotient rule
The quotient rule is a formula used in calculus to calculate the derivative of a function that is the quotient of two other functions
The quotient rule is a formula used in calculus to calculate the derivative of a function that is the quotient of two other functions. It is particularly useful when dealing with functions that cannot be simplified by factoring.
The formula for the quotient rule is as follows:
If f(x) = g(x) / h(x), then f'(x) = [g'(x) * h(x) – g(x) * h'(x)] / [h(x)]^2
To explain the quotient rule, let’s break it down step by step:
1. Start with the function f(x) that is the quotient of g(x) divided by h(x).
2. To find the derivative f'(x), we need to differentiate both the numerator (g(x)) and denominator (h(x)) separately.
3. Differentiate the numerator g(x) to find g'(x), the derivative of g(x).
4. Differentiate the denominator h(x) to find h'(x), the derivative of h(x).
5. Plug in the values of g'(x), h(x), g(x), and h'(x) into the quotient rule formula:
f'(x) = [g'(x) * h(x) – g(x) * h'(x)] / [h(x)]^2
6. Simplify the formula as much as possible by multiplying and combining terms.
7. Finally, the resulting expression is the derivative f'(x) of the original function f(x).
Here is an example to illustrate the quotient rule:
Let’s say we have the function f(x) = (x^2 + 1) / (2x – 1).
We want to find the derivative f'(x) using the quotient rule.
1. The function is f(x) = (x^2 + 1) / (2x – 1).
2. We need to differentiate the numerator and denominator separately.
g(x) = x^2 + 1
h(x) = 2x – 1
3. Differentiating the numerator:
g'(x) = 2x
4. Differentiating the denominator:
h'(x) = 2
5. Applying the quotient rule:
f'(x) = [g'(x) * h(x) – g(x) * h'(x)] / [h(x)]^2
= [2x * (2x – 1) – (x^2 + 1) * 2] / [(2x – 1)]^2
= (4x^2 – 2x – 2x^2 – 2) / (4x^2 – 4x + 1)
6. Simplifying the expression:
f'(x) = (2x^2 – 2x – 2) / (4x^2 – 4x + 1)
7. The resulting expression, (2x^2 – 2x – 2) / (4x^2 – 4x + 1), is the derivative f'(x) of the original function f(x).
So, f'(x) = (2x^2 – 2x – 2) / (4x^2 – 4x + 1) is the derivative of f(x) = (x^2 + 1) / (2x – 1) using the quotient rule.
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