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 u(x) and v(x), then the derivative of their product, denoted as (u*v)’ (read as “prime”), is given by the following formula:
(u*v)’ = u’ * v + u * v’
In other words, to find the derivative of a product, we take the derivative of one function (u’) and multiply it with the other function (v), then add it to the product of the first function (u) and the derivative of the second function (v’).
Let’s go through an example to understand how to apply the product rule.
Example:
Find the derivative of f(x) = x^2 * sin(x)
Solution:
To find the derivative of f(x), we need to use the product rule. Let’s identify our functions u(x) and v(x):
u(x) = x^2 (the function before the multiplication)
v(x) = sin(x) (the function after the multiplication)
Now, let’s find the derivatives of u(x) and v(x):
u'(x) = 2x (the derivative of x^2)
v'(x) = cos(x) (the derivative of sin(x))
Now, we can apply the product rule:
f'(x) = u'(x)*v(x) + u(x)*v'(x)
= (2x)*(sin(x)) + (x^2)*(cos(x))
This gives us the derivative of f(x), which is f'(x) = 2x*sin(x) + x^2*cos(x).
The product rule is a useful tool to find derivatives when dealing with functions that involve multiplication. Remembering and applying the formula correctly will help you solve more complex problems in calculus.
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