Product Rule
The product rule is a rule in calculus that allows us to find the derivative of a product of two functions
The product rule is a rule in calculus that allows us to find the derivative of a product of two functions. Suppose we have two functions, u(x) and v(x), and we want to find the derivative of their product, w(x) = u(x) * v(x).
The product rule states that the derivative of the product w(x) is given by:
w'(x) = u(x) * v'(x) + v(x) * u'(x)
In other words, the derivative of the product is equal to the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function.
To better understand and apply the product rule, let’s go through an example:
Example:
Find the derivative of f(x) = x^2 * sin(x)
Solution:
First, let’s identify u(x) and v(x) in our function. In this case, u(x) = x^2 and v(x) = sin(x).
Next, we need to find the derivatives of u(x) and v(x). The derivative of u(x) is u'(x) = 2x, and the derivative of v(x) is v'(x) = cos(x).
Now we can apply the product rule to find the derivative of f(x):
f'(x) = u(x) * v'(x) + v(x) * u'(x)
= x^2 * cos(x) + sin(x) * 2x
So the derivative of f(x) with respect to x is f'(x) = x^2 * cos(x) + sin(x) * 2x.
That’s how the product rule works. It allows us to find the derivative of a product of two functions by multiplying the first function by the derivative of the second function, and adding it to the second function multiplied by the derivative of the first function.
Remember to always apply the product rule when you have a function that is a product of two other functions.
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