The derivative of the function $x\arctan x$ is a common example encountered in calculus courses. This particular function showcases the application of the product rule and the derivative of the inverse tangent function. Understanding how to find this derivative is crucial for mastering differential calculus and its applications. Let's delve into the steps and reasoning behind finding the derivative of this function.
Finding the Derivative of $x\arctan x$
To differentiate $x\arctan x$, we'll employ the product rule. The product rule states that the derivative of the product of two functions, u(x) and v(x), is given by:
$ \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) $
In our case, $u(x) = x$ and $v(x) = \arctan x$. Let's find the derivatives of these individual functions:
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Derivative of u(x):
The derivative of $x$ is simply 1. Therefore, $u'(x) = 1$. -
Derivative of v(x): The derivative of the inverse tangent function, $\arctan x$, is given by: $ \frac{d}{dx}(\arctan x) = \frac{1}{1+x^2} $ So, $v'(x) = \frac{1}{1+x^2}$.
Now, we can substitute these derivatives into the product rule formula:
$ \begin{aligned} \frac{d}{dx}(x\arctan x) &= u'(x)v(x) + u(x)v'(x) \ &= (1)(\arctan x) + (x)\left(\frac{1}{1+x^2}\right) \ &= \arctan x + \frac{x}{1+x^2} \end{aligned} $
Therefore, the derivative of $x\arctan x$ is $\arctan x + \frac{x}{1+x^2}$.
Understanding the Steps
Let's break down the process to gain a deeper understanding:
- Identify the Functions: We recognized that $x\arctan x$ is a product of two functions, $x$ and $\arctan x$.
- Apply the Product Rule: We applied the product rule formula, which involves the derivatives of both functions and their product.
- Calculate Individual Derivatives: We found the derivative of each function, $x$ and $\arctan x$, separately.
- Substitute and Simplify: We plugged the individual derivatives back into the product rule formula and simplified the resulting expression.
Applications of the Derivative
The derivative of $x\arctan x$ has several applications in mathematics and related fields. Here are some key examples:
- Finding Critical Points: The derivative can be used to find critical points of a function. These are points where the derivative is equal to zero or undefined.
- Optimization Problems: Derivatives are essential for solving optimization problems, where we aim to find the maximum or minimum values of a function.
- Related Rates Problems: Derivatives are used to analyze the rate of change of quantities that are related to each other.
- Curve Sketching: The derivative helps to determine the increasing and decreasing intervals of a function, as well as its concavity.
Conclusion
Finding the derivative of $x\arctan x$ requires understanding the product rule and the derivative of the inverse tangent function. The process involves identifying the functions, applying the product rule, calculating individual derivatives, and simplifying the result. The derivative of this function has various applications in calculus and related fields, including finding critical points, solving optimization problems, and analyzing related rates. By mastering this concept, you gain a valuable tool for solving a wide range of mathematical problems.
