quotient rule
g(x)f'(x)-f(x)g'(x)/g(x)^2
Quotient rule is a formula used in calculus to find the derivative of the quotient of two functions, which represents the rate at which the output of the function changes with respect to the input. The quotient rule is particularly useful in cases where we have to find the derivative of a complex function that involves more than one operation.
The quotient rule states that the derivative of the quotient of two functions u(x) and v(x) is given by:
(f/g)’ = [(u’v) – (uv’)]/v^2
Where u’ and v’ are the derivatives of the functions u and v with respect to x, respectively.
To apply the quotient rule, follow these steps:
1. Identify the numerator and the denominator of the quotient function.
2. Apply the product rule to the numerator, which yields (u’v)+(uv’).
3. Apply the chain rule to the denominator, finding the derivative of the denominator as (v^2)’. This yields 2v(v’).
4. Substitute the values of u’, v’, and v into the formula (f/g)’ = [(u’v) – (uv’)]/v^2.
5. Simplify the expression algebraically to obtain the derivative of the original function.
In summary, the quotient rule is an essential tool in calculus to calculate the derivative of a quotient function. It can help to find the rate of change of a function and is useful in complex calculus problems.
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