The Product Rule for Derivatives¶

Before we delve into the product rule, let's first understand what it means to differentiate a function of one variable. When we take the derivative of a function, f(x), we are finding the slope of its tangent line at any given point x. This is essentially calculating how the function changes as we move along the x-axis. In other words, it tells us the instantaneous rate of change of the function.

Why (fg)' ≠ f'g'?¶

Consider two simple functions: f(x) = x^2 and g(x) = sin(x). If we multiply these two functions together, we get a new function h(x) = x^2 * sin(x). Now, let's try to find the derivative of h(x) by simply taking the product of the derivatives: (f(x))' and (g(x))'.

If we do this, we get:

h'(x) = (x^2)' * (sin(x))' = 2x * cos(x), which is not the correct derivative of h(x). The correct derivative of h(x) would be (x^2 * sin(x))'.

Why did our intuition fail us here? To understand this, let's dive into the product rule.

The Product Rule for Derivatives¶

The product rule is a powerful tool in calculus that allows us to find the derivative of a function that is the product of two other functions. It states:

(fg)' = f'g + fg'

This rule tells us that we cannot simply multiply the derivatives of the individual functions and expect to obtain the correct derivative of their product. Instead, we must apply the product rule to correctly find the derivative of the product function. Let's see this in action with a few examples.

Example 1: Product of Polynomials¶

Let's take two polynomials: f(x) = x^2 and g(x) = x^3. We want to find the derivative of their product, h(x) = x^5. Using the product rule, we get:

h'(x) = (f(x))' * (g(x)) + (f(x)) * (g(x))' = (2x) * (3x^2) + (x^2) * (3x^2) = 6x^3 + 3x^4

So, the derivative of h(x) is 6x^3 + 3x^4.

Example 2: Product of Polynomial and Exponential Function¶

Let's take a polynomial f(x) = x^2 and an exponential function g(x) = e^x. We want to find the derivative of their product, h(x) = x^2 * e^x. Using the product rule, we get:

h'(x) = (f(x))' * (g(x)) + (f(x)) * (g(x))' = (2x) * (e^x) + (x^2) * (e^x) = 2x * e^x + x^2 * e^x

So, the derivative of h(x) is 2x * e^x + x^2 * e^x.

Example 3: Product of Polynomial and Trigonometric Function¶

Let's take a polynomial f(x) = x^2 and a trigonometric function g(x) = sin(x). We want to find the derivative of their product, h(x) = x^2 * sin(x). Using the product rule, we get:

h'(x) = (f(x))' * (g(x)) + (f(x)) * (g(x))' = (2x) * (sin(x)) + (x^2) * (cos(x)) = 2x * sin(x) + x^2 * cos(x)

So, the derivative of h(x) is 2x * sin(x) + x^2 * cos(x).

Common Mistakes¶

1. Ignoring the Product Rule: The product rule should be applied whenever we have a function that is the product of two other functions. Failing to apply the product rule will result in an incorrect derivative.

2. Confusing Differentiation and Integration: While both differentiation and integration involve manipulating functions, they serve entirely different purposes. Confusing these concepts can lead to errors when applying the product rule or any other calculus rules.

3. Neglecting Constants: When applying the product rule, make sure to consider any constant factors within each of the functions being multiplied. These constants should be included in both the initial product and its derivative.

4. Incorrect Application of Rules: The product rule is only one of many rules used in calculus. Ensuring that you apply the correct rule for a given situation will avoid unnecessary mistakes.

Conclusion¶

The product rule is an essential tool in calculus, enabling us to find the derivative of a function that is the product of two other functions. By understanding this rule and applying it correctly, we can tackle more complex mathematical problems and better understand the world around us. Remember to always check your work and be mindful of common pitfalls when using rules like the product rule.