The Product Rule in Calculus: How to Find the Derivative of a Product of Two Functions

Product Rule

The product rule is a formula used in calculus to find the derivative of the product of two functions

The product rule is a formula used in calculus to find the derivative of the product of two functions. It is often used when we need to differentiate functions that are multiplied together. The product rule states that if we have two functions, f(x) and g(x), their product’s derivative can be found using the following formula:

d/dx [f(x) * g(x)] = f'(x) * g(x) + f(x) * g'(x)

In simpler terms, to differentiate the product of two functions, we take the derivative of the first function and multiply it by the second function, then we add that to the product of the first function and the derivative of the second function.

Let’s go through an example to understand how to apply the product rule.

Example: Find the derivative of the function h(x) = (2x^2 + 3x) * (4x^3 – 6x)

To use the product rule, we need to identify the two functions that are being multiplied together. In this case, the two functions are f(x) = 2x^2 + 3x and g(x) = 4x^3 – 6x.

Now, we can differentiate both functions separately to find their derivatives:

f'(x) = d/dx (2x^2 + 3x)
= 4x + 3

g'(x) = d/dx (4x^3 – 6x)
= 12x^2 – 6

Next, we apply the product rule by using the formula:

d/dx [f(x) * g(x)] = f'(x) * g(x) + f(x) * g'(x)

Plugging in the values, we have:

h'(x) = (4x + 3) * (4x^3 – 6x) + (2x^2 + 3x) * (12x^2 – 6)

Now we can simplify the expression:

h'(x) = (16x^4 – 24x^2 + 12x^3 – 18x) + (24x^4 + 36x^3 – 12x^2 – 18x)

Combining like terms, we get:

h'(x) = 40x^4 + 48x^3 – 36x^2 – 36x

So, the derivative of h(x) = (2x^2 + 3x) * (4x^3 – 6x) is h'(x) = 40x^4 + 48x^3 – 36x^2 – 36x.

Remember to always check your work and simplify if possible. Practice using the product rule with various functions to gain confidence in applying this formula.

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