quotient rule
The quotient rule is a formula used to find the derivative of a function that is in the form of a quotient, where one function is divided by another function
The quotient rule is a formula used to find the derivative of a function that is in the form of a quotient, where one function is divided by another function. It is a key rule in calculus and is used to differentiate a wide range of functions.
The formula for the quotient rule is as follows:
If we have a function f(x) = g(x) / h(x), where g(x) and h(x) are both differentiable functions, then the derivative of f(x) is given by:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2
Let’s break down the formula step by step:
1. Find the derivative of the numerator: g'(x)
This represents the derivative of the function in the numerator.
2. Multiply the numerator derivative by the denominator function: g'(x) * h(x)
This step involves taking the derivative of g(x) and then multiplying it by h(x).
3. Find the derivative of the denominator: h'(x)
This represents the derivative of the function in the denominator.
4. Multiply the denominator derivative by the numerator function: g(x) * h'(x)
This step involves taking the derivative of h(x) and then multiplying it by g(x).
5. Subtract the result in step 4 from the result in step 2: g'(x) * h(x) – g(x) * h'(x)
This equation represents the numerator of the derivative.
6. Square the denominator: (h(x))^2
This step is necessary to complete the denominator of the derivative.
7. Divide the result from step 5 by the result from step 6: (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2
This equation gives the derivative of the function f(x).
By applying the quotient rule, we can differentiate complex functions that involve the division of two functions. It is important to remember to simplify and combine terms if possible after applying the rule.
It is worth noting that the quotient rule is derived from the product rule, which is another fundamental rule in calculus. The product rule states that if we have a function f(x) = g(x) * h(x), then the derivative of f(x) is given by:
f'(x) = g'(x) * h(x) + g(x) * h'(x)
The difference in the quotient rule is that we subtract rather than add the two terms, and divide by the square of the denominator.
More Answers:
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