Question
quotient rule example
Answers
GPT 3.5
The quotient rule is used to find the derivative of a function that is the quotient of two other functions. Mathematically, it can be stated as:
If f(x) = g(x) / h(x), then f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2
Here's an example to illustrate the quotient rule:
Let's find the derivative of f(x) = x^2 / (2x + 1).
Using the quotient rule, we have:
f'(x) = [(2x + 1)(2x) - (x^2)(2)] / (2x + 1)^2
Simplifying the numerator:
f'(x) = (4x^2 + 2x - 2x^2) / (2x + 1)^2
Combining like terms:
f'(x) = (2x^2 + 2x) / (2x + 1)^2
Simplifying further is not possible in this case. So, that's the derivative of f(x) using the quotient rule.
If f(x) = g(x) / h(x), then f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2
Here's an example to illustrate the quotient rule:
Let's find the derivative of f(x) = x^2 / (2x + 1).
Using the quotient rule, we have:
f'(x) = [(2x + 1)(2x) - (x^2)(2)] / (2x + 1)^2
Simplifying the numerator:
f'(x) = (4x^2 + 2x - 2x^2) / (2x + 1)^2
Combining like terms:
f'(x) = (2x^2 + 2x) / (2x + 1)^2
Simplifying further is not possible in this case. So, that's the derivative of f(x) using the quotient rule.