The U by V rule, more commonly known as the product rule in calculus, describes how to differentiate the product of two functions.
Understanding the Product Rule
The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Mathematically:
- d(uv)/dx = (du/dx)v + u(dv/dx)
Where:
- u and v are both differentiable functions of x.
- du/dx represents the derivative of u with respect to x.
- dv/dx represents the derivative of v with respect to x.
This rule is crucial for differentiating functions that are expressed as products. Simply differentiating each part individually and multiplying the results is incorrect.
Examples
Let's illustrate with some examples:
-
Find the derivative of f(x) = x²sin(x).
Here, u = x² and v = sin(x). Therefore:
- du/dx = 2x
- dv/dx = cos(x)
Applying the product rule:
- f'(x) = (2x)sin(x) + x²(cos(x)) = 2xsin(x) + x²cos(x)
-
Find the derivative of g(x) = (3x + 2)(x² - 1).
Here, u = 3x + 2 and v = x² - 1. Therefore:
- du/dx = 3
- dv/dx = 2x
Applying the product rule:
- g'(x) = 3(x² - 1) + (3x + 2)(2x) = 3x² - 3 + 6x² + 4x = 9x² + 4x - 3
Practical Insights
The product rule is fundamental in various applications of calculus, including:
- Physics: Calculating rates of change involving multiple interacting factors.
- Engineering: Analyzing systems with multiple components whose behavior affects one another.
- Economics: Modeling the impact of changes in multiple variables on a dependent variable.
The product rule provides a systematic way to handle the differentiation of complex functions, simplifying calculations and providing accurate results.
