Understanding the Derivative of ( x \arctan x )
The function ( f(x) = x \arctan x ) is a product of two functions: ( u(x) = x ) and ( v(x) = \arctan x ). To find its derivative, we apply the product rule, which states:
[ (uv)’ = u’v + uv’ ]
Step 1: Compute the derivatives of ( u(x) ) and ( v(x) )
- ( u(x) = x ) implies ( u’(x) = 1 ).
- ( v(x) = \arctan x ). To find ( v’(x) ), recall the derivative of ( \arctan x ) is ( \frac{1}{1 + x^2} ).
Step 2: Apply the product rule
[ \frac{d}{dx}(x \arctan x) = (1)(\arctan x) + (x)\left(\frac{1}{1 + x^2}\right) ]
Step 3: Simplify the expression
[ \frac{d}{dx}(x \arctan x) = \arctan x + \frac{x}{1 + x^2} ]
Final Result
[ \boxed{\frac{d}{dx}(x \arctan x) = \arctan x + \frac{x}{1 + x^2}} ]
The derivative of x \arctan x combines the product rule with the known derivative of \arctan x , resulting in a sum of \arctan x and a rational function.
Why This Matters
Understanding the derivative of ( x \arctan x ) is crucial in calculus for:
- Integration techniques: It aids in solving integrals involving ( \arctan x ) through substitution or parts.
- Optimization problems: Derivatives help identify critical points of functions involving inverse trigonometric terms.
- Modeling: Functions like ( x \arctan x ) appear in physics, engineering, and economics, where their rates of change are essential.
The product rule is a cornerstone of differential calculus, enabling the differentiation of composite functions. Combining it with inverse trigonometric derivatives expands its utility in advanced mathematical analysis.
Historical Context
The arctangent function emerged in the 17th century as mathematicians like Leibniz and Euler explored inverse trigonometric relationships. Its derivative, ( \frac{1}{1 + x^2} ), was derived from geometric and algebraic principles, laying the groundwork for modern calculus.
Practical Application
Consider a scenario in physics where ( x \arctan x ) models the displacement of a damped harmonic oscillator. Its derivative represents the velocity, critical for analyzing system dynamics.
Common Misconceptions
Myth: The derivative of x \arctan x can be found using the chain rule alone.
Reality: The product rule is necessary because x \arctan x is a product of two functions, not a composite function.
FAQ Section
What is the derivative of \arctan x alone?
+The derivative of \arctan x is \frac{1}{1 + x^2} , derived from the definition of inverse functions and trigonometric identities.
Can the product rule be avoided in this derivation?
+No, the product rule is essential here since x \arctan x is a product of two functions. Other rules like the chain rule do not apply directly.
How is x \arctan x used in real-world applications?
+It appears in modeling signals, electrical circuits, and mechanical systems where nonlinear relationships need to be described.
Future Trends
As computational methods advance, functions like ( x \arctan x ) and their derivatives become integral to machine learning algorithms, particularly in neural networks where activation functions resemble inverse trigonometric forms.
Conclusion
The derivative of ( x \arctan x ) exemplifies the interplay between fundamental calculus rules and special functions. Mastery of such derivations not only strengthens mathematical skills but also empowers problem-solving across scientific and engineering disciplines.
