product rule
The product rule is a rule in calculus that allows you to find the derivative of a function that is the product of two other functions
The product rule is a rule in calculus that allows you to find the derivative of a function that is the product of two other functions. It is often used when dealing with functions that cannot be simplified or differentiated easily.
Mathematically, the product rule states that if you have two functions f(x) and g(x), the derivative of their product, denoted as (f(x) * g(x))’, can be found using the following formula:
(f(x) * g(x))’ = f'(x) * g(x) + f(x) * g'(x)
To understand and use the product rule, follow these steps:
1. Identify the functions f(x) and g(x) in the given problem.
2. Find the derivatives of both functions, f'(x) and g'(x), individually.
3. Apply the product rule formula, plugging in the expressions from step 2:
(f(x) * g(x))’ = f'(x) * g(x) + f(x) * g'(x)
4. Simplify the expression if possible.
Let’s illustrate the product rule with an example:
Example:
Find the derivative of the function f(x) = x^2 * sin(x).
Solution:
1. Identify f(x) = x^2 and g(x) = sin(x).
2. Find f'(x) and g'(x):
– f'(x) = 2x (using the power rule)
– g'(x) = cos(x) (derivative of sin(x) is cos(x))
3. Apply the product rule formula:
(f(x) * g(x))’ = (2x * sin(x)) + (x^2 * cos(x))
4. Simplify the expression if necessary.
So, the derivative of f(x) = x^2 * sin(x) is f'(x) = 2x * sin(x) + x^2 * cos(x).
Remember, the product rule is an essential tool when dealing with functions in calculus that involve multiplication. It helps us find the derivative without resorting to more complicated methods.
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