Derivative of Product of Two Functions

When dealing with calculus, one often encounters the situation where it is necessary to take the derivative of the product of two functions. This is not as straightforward as taking the derivative of a single function, and it requires the use of a specific rule known as the Product Rule.

The Product Rule

The Product Rule is a formula used to find the derivative of a product of two functions. It states that if you have two differentiable functions, $f(x)$ and $g(x)$, then the derivative of their product $h(x) = f(x)g(x)$ is given by:

$$ h'(x) = f'(x)g(x) + f(x)g'(x) $$

This rule is essential because the derivative of a product is not simply the product of the derivatives. The Product Rule accounts for the interaction between the two functions as they change.

Understanding the Product Rule

To understand why the Product Rule works, consider the following:

• When one function increases, the product increases.
• When the other function increases, the product also increases.
• The total rate of change of the product depends on both of these increases.

The Product Rule combines these effects by adding the rate of change of $f(x)$ times the value of $g(x)$ to the rate of change of $g(x)$ times the value of $f(x)$.

Table of Differences and Important Points

Aspect	Derivative of Single Function	Derivative of Product of Two Functions
Rule	Power Rule, Chain Rule, etc.	Product Rule
Formula	$f'(x)$	$f'(x)g(x) + f(x)g'(x)$
Complexity	Simple	More complex due to interaction
Application	Direct differentiation	Requires consideration of both functions
Example	$\frac{d}{dx}(x^2) = 2x$	$\frac{d}{dx}(x^2 \cdot \sin(x)) = 2x \cdot \sin(x) + x^2 \cdot \cos(x)$

Examples

Example 1: Basic Functions

Let's take the functions $f(x) = x^2$ and $g(x) = \sin(x)$. To find the derivative of their product $h(x) = x^2 \cdot \sin(x)$, we apply the Product Rule:

$$ \begin{align*} h'(x) &= \frac{d}{dx}(x^2) \cdot \sin(x) + x^2 \cdot \frac{d}{dx}(\sin(x)) \ &= 2x \cdot \sin(x) + x^2 \cdot \cos(x) \end{align*} $$

Example 2: Exponential and Trigonometric Functions

Consider $f(x) = e^x$ and $g(x) = \cos(x)$. The derivative of $h(x) = e^x \cdot \cos(x)$ is:

$$ \begin{align*} h'(x) &= \frac{d}{dx}(e^x) \cdot \cos(x) + e^x \cdot \frac{d}{dx}(\cos(x)) \ &= e^x \cdot \cos(x) - e^x \cdot \sin(x) \end{align*} $$

Example 3: Polynomial and Logarithmic Functions

For $f(x) = x^3$ and $g(x) = \ln(x)$, the derivative of $h(x) = x^3 \cdot \ln(x)$ is found as follows:

$$ \begin{align*} h'(x) &= \frac{d}{dx}(x^3) \cdot \ln(x) + x^3 \cdot \frac{d}{dx}(\ln(x)) \ &= 3x^2 \cdot \ln(x) + x^3 \cdot \frac{1}{x} \ &= 3x^2 \cdot \ln(x) + x^2 \end{align*} $$

Conclusion

The Product Rule is a fundamental tool in calculus for finding the derivative of the product of two functions. It is crucial to remember that you cannot simply multiply the derivatives of the individual functions; instead, you must apply the Product Rule to correctly account for the interaction between the functions. Understanding and applying this rule is essential for solving complex problems in calculus.