Post 3: Product and Quotient Rules

In calculus, we often encounter functions that are products or quotients of other functions. The product and quotient rules are two powerful tools that enable us to find the derivatives of these types of functions. Let's explore these rules in more detail.

Product Rule:

The product rule allows us to find the derivative of a product of two functions. Let's say we have two functions, f(x) and g(x), whose derivatives we want to find. The product rule states:

In other words, to find the derivative of a product, we differentiate the first function and multiply it by the second function, then add that to the product of the first function and the derivative of the second function.

Example 1: Let's find the derivative of the function h(x) = x^2 * sin(x).

Using the product rule, we have: [ \frac{{d}}{{dx}} \left(x^2 \cdot \sin(x)\right) = 2x \cdot \sin(x) + x^2 \cdot \cos(x) ]

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

Quotient Rule:

The quotient rule helps us find the derivative of a quotient of two functions. Suppose we have two functions, f(x) and g(x), and we want to find the derivative of their quotient f(x)/g(x). The quotient rule can be stated as follows:

To apply the quotient rule, we differentiate the numerator, multiply it by the denominator, then subtract the product of the numerator and the derivative of the denominator, all divided by the square of the denominator.

Example 2: Let's find the derivative of the function r(x) = (3x^2 + 2x) / x.

Using the quotient rule, we have: [ \frac{{d}}{{dx}} \left(\frac{{3x^2 + 2x}}{{x}}\right) = \frac{{(6x + 2) \cdot x - (3x^2 + 2x) \cdot 1}}{{x^2}} ] Simplifying this expression, we get: [ \frac{{6x^2 + 2x - 3x^2 - 2x}}{{x^2}} = \frac{{3x^2}}{{x^2}} = 3 ]

Therefore, the derivative of r(x) is 3.

By using the product and quotient rules, we can easily find the derivatives of more complex functions that involve products or quotients of multiple functions. These rules are invaluable tools in calculus and allow us to analyze and understand the behavior of various mathematical functions.