Mastering Simplification- Harnessing the Product Rule’s Power for Efficient Calculations

Which applies the power of a product rule to simplify

In mathematics, the product rule is a fundamental concept in calculus that allows us to differentiate the product of two functions. This rule is particularly useful when dealing with complex expressions, as it simplifies the process of finding derivatives. One of the most effective ways to apply the power of the product rule to simplify an expression is by using it to differentiate the product of two functions. In this article, we will explore the various applications of the product rule and how it can be used to simplify different types of functions.

The product rule states that if we have two functions, f(x) and g(x), then the derivative of their product, f(x)g(x), is given by:

(fg)'(x) = f'(x)g(x) + f(x)g'(x)

This rule can be applied to simplify a wide range of expressions, including polynomials, trigonometric functions, and exponential functions. Let’s take a look at some examples to illustrate how the product rule can be used to simplify these types of functions.

Consider the following expression:

(h(x))^2 = (x^2 + 2x + 1)^2

To simplify this expression, we can apply the power of the product rule by differentiating both sides with respect to x:

2(h(x))h'(x) = 2(x^2 + 2x + 1)(2x + 2)

Now, we can simplify the right-hand side of the equation by applying the product rule to the two factors:

2(h(x))h'(x) = 4x^3 + 8x^2 + 4x + 4

Dividing both sides by 2, we get:

h(x)h'(x) = 2x^3 + 4x^2 + 2x + 2

This simplified expression can now be used to find the derivative of h(x) with respect to x, or to solve various problems involving the function.

Another example involves the product of two trigonometric functions:

sin(x)cos(x)

To simplify this expression, we can use the product rule to differentiate both sides with respect to x:

(cos(x))(cos(x)) – (sin(x))(sin(x)) = cos^2(x) – sin^2(x)

This simplified expression is known as the Pythagorean identity, which is a fundamental trigonometric relation. By applying the power of the product rule, we were able to derive a useful identity that can be used to solve various trigonometric problems.

In conclusion, the power of the product rule is a valuable tool in calculus that can be used to simplify a wide range of expressions. By applying this rule, we can differentiate the product of two functions and derive useful identities that can be used to solve complex problems. Whether you are dealing with polynomials, trigonometric functions, or exponential functions, the product rule is a powerful tool that can help you simplify and solve mathematical expressions more efficiently.