derivative of x 2cosx

2 min read 21-10-2024

Unraveling the Derivative of x²cos(x): A Step-by-Step Guide

The derivative of a function tells us the rate at which it changes. In this article, we'll explore the derivative of the function x²cos(x). This function combines a simple polynomial term (x²) with a trigonometric term (cos(x)), making its derivative a bit more intricate. We'll break down the process step by step, making it accessible for anyone with a basic understanding of calculus.

Understanding the Product Rule

The core concept we'll use is the product rule of differentiation. This rule states that the derivative of a product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function.

Mathematically:

d/dx [u(x) * v(x)] = u(x) * v'(x) + v(x) * u'(x)

Where u(x) and v(x) are differentiable functions.

Applying the Product Rule to x²cos(x)

Let's apply this rule to our function x²cos(x).

Identify the functions:
- u(x) = x²
- v(x) = cos(x)
Find the derivatives:
- u'(x) = 2x (derivative of x²)
- v'(x) = -sin(x) (derivative of cos(x))
Apply the product rule:
- d/dx [x²cos(x)] = x² * (-sin(x)) + cos(x) * 2x
Simplify the expression:
- d/dx [x²cos(x)] = -x²sin(x) + 2xcos(x)

Therefore, the derivative of x²cos(x) is -x²sin(x) + 2xcos(x).

Visualizing the Derivative

Let's visualize the original function x²cos(x) and its derivative * -x²sin(x) + 2xcos(x)*. We can see how the derivative captures the rate of change of the original function at every point.

[Insert graph of x²cos(x) and its derivative]

Note: The derivative of a function can be positive, negative, or zero. A positive derivative indicates that the original function is increasing, a negative derivative indicates a decreasing function, and a zero derivative signifies a stationary point (where the function is neither increasing nor decreasing).

Applications of the Derivative

The derivative of a function has various applications in mathematics, physics, and other fields:

Finding the slope of a tangent line: The derivative at a point on a curve gives the slope of the tangent line to that point.
Finding the maximum and minimum values of a function: The derivative can be used to locate critical points where the function may have maximum or minimum values.
Modeling real-world phenomena: Derivatives are essential for modeling physical systems and their behavior, such as the motion of objects or the rate of change of temperature.

Conclusion

Understanding the derivative of a function, especially a combination of different types of functions like x²cos(x), is crucial in various areas of study. The product rule, as demonstrated here, is a powerful tool that allows us to calculate the derivative of complex functions. By applying the product rule and understanding its implications, we can gain valuable insights into the behavior of functions and their applications in the real world.