2.8 The Product Rule

The Product Rule is a rule in Calculus that states that the derivative of the product of two functions is equal to the derivative of the first function times the second function plus the first function times the derivative of the second function. This rule is represented mathematically as: (f(x) * g(x))' = f'(x) * g(x) + f(x) * g'(x) This rule allows us to find the derivative of a function that is the product of two other functions. For example, if we have a function f(x) = x^2 and a function g(x) = 3x, we can use the Product Rule to find the derivative of the function h(x) = f(x) * g(x) = x^2 * 3x = 3x^3. The derivative of h(x) = f'(x) * g(x) + f(x) * g'(x) = 2x * 3x + x^2 * 3 = 6x^2 + 3x^2 = 9x^2. Another example, Find the derivative of f(x)= (x^2+5x+6) * (2x^2+x) Solution: Using the product rule, the derivative of f(x) = (x^2+5x+6) * (2x^2+x) = (x^2+5x+6)' * (2x^2+x) + (x^2+5x+6) * (2x^2+x)' = (2x+5) * (2x^2+x) + (x^2+5x+6) * (4x+1) = 2x(2x^2+x) + 5(2x^2+x) + (x^2+5x+6)(4x+1) = 4x^3+2x^2+5x^3+5x^2+4x^3+4x^2+6x = 9x^3 + 9x^2 + 6x It's important to note that the product rule can also be used in combination with other rules, such as the Chain rule, to find the derivatives of more complex functions.

Example 1: Find the derivative of f(x) = x^3 * sin x Solution: Using the product rule, the derivative of f(x) = x^3 * sin x = (x^3)' * sin x + x^3 * (sin x)' = 3x^2 * sin x + x^3 * cos x Example 2: Find the derivative of f(x) = e^x * ln x Solution: Using the product rule, the derivative of f(x) = e^x * ln x = (e^x)' * ln x + e^x * (ln x)' = e^x * ln x + e^x * (1/x) Example 3: Find the derivative of f(x) = (x^2+5x+6) * (2x^2+x) Solution: Using the product rule, the derivative of f(x) = (x^2+5x+6) * (2x^2+x) = (2x+5) * (2x^2+x) + (x^2+5x+6) * (4x+1) = 9x^3 + 9x^2 + 6x

Example 4:

Find the derivative of f(x) = x^4 * cos x

Solution:

Using the product rule, the derivative of f(x) = x^4 * cos x = (x^4)' * cos x + x^4 * (cos x)' = 4x^3 * cos x - x^4 * sin x

Example 5:

Find the derivative of f(x) = (x^2 + 3x + 2) * (e^x + x^2)

Solution:

Using the product rule, the derivative of f(x) = (x^2 + 3x + 2) * (e^x + x^2) = (2x + 3) * (e^x + x^2) + (x^2 + 3x + 2) * (e^x + 2x)

Example 6:

Find the derivative of f(x) = sin(x^2) * ln(x+2)

Solution:

Using the product rule, the derivative of f(x) = sin(x^2) * ln(x+2) = cos(x^2) * 2x * ln(x+2) + sin(x^2) * (1/(x+2))

In these examples, we can see how the Product Rule can be applied to different types of functions and how it can be used in combination with other rules to find the derivatives of more complex functions. It is important to note that the Product Rule applies to any combination of functions where one function is being multiplied by another, not just the examples shown here. As always, practice and understanding the basic rules of Calculus will help make solving problems like these easier.