Product Rule:

The product rule is a fundamental rule in calculus that helps us differentiate a function that is a product of two or more functions. It states that if we have a function $h(x) = f(x) \cdot g(x)$, where $f(x)$ and $g(x)$ are both differentiable functions, then the derivative of $h(x)$ with respect to $x$ is given by:

In simpler terms, the product rule tells us that the derivative of a product of functions is equal to the derivative of one function times the other function plus the derivative of the second function times the first function.

This rule is particularly useful when dealing with composite functions, which are functions made up of multiple functions combined together. By applying the product rule repeatedly, we can find the derivative of even the most complex functions.

Now, let's see how the product rule works in practice. Let's consider the function:

where $f(x) = x^2 - 2$ and $g(x) = x^3 - 5x^2 + 6$. We want to find the derivative of $h(x)$ using the product rule. First, we take the derivative of each function separately:

Then, we apply the product rule:

Substituting the derivatives we just found, we get:

Expanding the expressions and simplifying, we arrive at:

And there you have it! The derivative of $h(x)$ has been successfully found using the product rule. With this powerful tool, we can tackle much more complex functions than this simple example. Remember, whenever you encounter a product of functions, just break out your trusty product rule and follow the formula. Differentiation will become second nature in no time!