The Product Rule: How to Find the Derivative of the Multiplication of Two Functions

derivative of the multiplaction of two functions

f(x)g'(x) + g(x)f'(x)

To find the derivative of the multiplication of two functions, you can use the product rule. The product rule states that if you have two functions, let’s call them u(x) and v(x), then the derivative of their product, which we will denote as w(x) = u(x) * v(x), can be calculated as follows:

w'(x) = u'(x) * v(x) + u(x) * v'(x)

Here, u'(x) represents the derivative of u(x) with respect to x, and v'(x) represents the derivative of v(x) with respect to x.

Let’s demonstrate the use of the product rule with an example. Suppose we have two functions, f(x) = 3x^2 and g(x) = 2x. We want to find the derivative of their product h(x) = f(x) * g(x).

First, we differentiate f(x) and g(x) separately. Taking the derivative of f(x) and g(x) gives:

f'(x) = 6x
g'(x) = 2

Applying the product rule, we can find the derivative of h(x):

h'(x) = f'(x) * g(x) + f(x) * g'(x)
= (6x) * (2x) + (3x^2) * (2)
= 12x^2 + 6x^2
= 18x^2

So, the derivative of h(x) is 18x^2.

In summary, to find the derivative of the multiplication of two functions, apply the product rule by multiplying the derivative of one function with the other function and vice versa, and then sum these two products together.

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