Derivative of arctan x

Definition

The derivative of arctan x refers to the rate of change of the inverse tangent function with respect to its argument, x. It is a crucial concept in understanding how inverse trigonometric functions behave and is derived using implicit differentiation or geometric interpretations, particularly in the context of right triangles and the unit circle. This derivative is important for solving problems related to optimization, motion, and rates of change.

5 Must Know Facts For Your Next Test

1. The derivative of arctan x is given by the formula $$\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$$, valid for all real values of x.
2. The graph of the derivative $$\frac{1}{1+x^2}$$ indicates that it is always positive, meaning that arctan x is an increasing function across its entire domain.
3. As x approaches positive or negative infinity, the arctan x approaches $$\frac{\pi}{2}$$ and $$-\frac{\pi}{2}$$ respectively, which reflects its horizontal asymptotes.
4. The derivative helps to understand the slope of the arctan curve at any point, providing insights into how steep or flat the curve is.
5. The derivative's value decreases as |x| increases, showing that while arctan x continues to increase, it does so at a decreasing rate.

Review Questions

• How can you derive the formula for the derivative of arctan x using implicit differentiation?
  • To derive the formula for the derivative of arctan x using implicit differentiation, start with the equation $$y = \arctan x$$. This implies that $$x = \tan y$$. Taking the derivative of both sides with respect to x gives us $$1 = \sec^2 y \cdot \frac{dy}{dx}$$. Since $$\sec^2 y = 1 + \tan^2 y$$ and we can substitute back $$\tan y = x$$, we find that $$\sec^2 y = 1 + x^2$$. Thus, rearranging gives us $$\frac{dy}{dx} = \frac{1}{1 + x^2}$$.
• Discuss the significance of the positive nature of the derivative of arctan x in relation to its graphical representation.
  • The fact that the derivative of arctan x, given by $$\frac{1}{1+x^2}$$, is always positive means that the function itself is increasing for all real numbers. In terms of its graphical representation, this implies that as you move along the x-axis in either direction, the value of arctan x continues to rise. This characteristic allows us to understand not only how the function behaves but also reinforces its role as an inverse function to tangent, where each output corresponds uniquely to an input.
• Evaluate how understanding the derivative of arctan x can assist in solving complex real-world problems involving angles and slopes.
  • Understanding the derivative of arctan x allows one to apply concepts from calculus to a variety of real-world situations involving angles and slopes. For instance, in physics, this derivative can help determine the angle of elevation or depression in projectile motion problems or when analyzing inclined planes. By knowing how quickly angles change relative to distances (or other parameters), one can better predict behaviors and outcomes in fields such as engineering and navigation. This comprehension elevates analytical skills and improves decision-making based on mathematical principles.

Related terms

arctangent function: The arctangent function, denoted as arctan or tan^{-1}, is the inverse of the tangent function and gives the angle whose tangent is a given number.

chain rule: The chain rule is a fundamental technique in calculus used to find the derivative of composite functions, essential for differentiating functions like arctan that are often combined with other expressions.

trigonometric identities: Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables; they are often used in conjunction with derivatives to simplify expressions.