How to Apply the Product Rule in Calculus to Differentiate Functions

Product Rule

The product rule is a basic rule in calculus that allows us to differentiate the product of two functions

The product rule is a basic rule in calculus that allows us to differentiate the product of two functions. It is used when we have a function that is a multiple of two other functions, and we want to find the derivative of this combined function.

The product rule states that if we have two functions, let’s call them u(x) and v(x), then the derivative of their product is given by:

(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)

In other words, to differentiate the product of two functions, we differentiate the first function and multiply it by the second function, then we add the original first function multiplied by the derivative of the second function.

To better understand and apply the product rule, let’s go through an example:

Example: Find the derivative of the function f(x) = x^2 * sin(x)

To find the derivative of this function, we will use the product rule. Let’s assign u(x) = x^2 and v(x) = sin(x).

According to the product rule, the derivative of f(x) is given by:

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

Now, let’s find the derivatives of u(x) and v(x):

u'(x) = d/dx (x^2) = 2x

v'(x) = d/dx (sin(x)) = cos(x)

Now, we can substitute these derivatives back into our equation:

f'(x) = u'(x) * v(x) + u(x) * v'(x)
= (2x) * sin(x) + (x^2) * cos(x)

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 a fundamental tool when dealing with functions that are the product of two other functions. By applying this rule, we can find the derivative of such functions efficiently.

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