Product Rule
The product rule is a formula used in calculus to find the derivative of a product of two functions
The product rule is a formula used in calculus to find the derivative of a product of two functions. It states that if you have two functions, let’s call them f(x) and g(x), the derivative of their product, denoted as (f(x) * g(x)), is given by:
(f(x) * g(x))’ = f'(x) * g(x) + f(x) * g'(x)
In words, the product rule tells us that to find the derivative of a product of two functions, we take the derivative of the first function multiplied by the second function, and then add it to the product of the first function with the derivative of the second function.
To better understand the product rule, let’s consider an example. Suppose we have two functions:
f(x) = x^2 (function 1)
g(x) = cos(x) (function 2)
To find the derivative of their product, (f(x) * g(x)), we can use the product rule. Let’s apply it step-by-step:
1. Find the derivative of the first function, f'(x):
f'(x) = 2x
2. Find the derivative of the second function, g'(x):
g'(x) = -sin(x)
3. Apply the product rule:
(f(x) * g(x))’ = f'(x) * g(x) + f(x) * g'(x)
(x^2 * cos(x))’ = 2x*cos(x) + x^2*(-sin(x))
= 2x*cos(x) – x^2*sin(x)
So, the derivative of the product (x^2 * cos(x)) is 2x*cos(x) – x^2*sin(x).
The product rule is a fundamental tool in calculus that allows us to differentiate products of functions. It is important to remember and apply this rule when dealing with problems where functions are multiplied together.
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