Finding the Derivative of tan inverse x : Step-by-Step Guide

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The concept of differentiation is a fundamental pillar of calculus, allowing us to understand how functions change and giving us insights into their behavior. Among the various functions encountered in calculus, the arctangent function, often denoted as tan⁻¹(x) or arctan(x), holds a special place. This article delves into the process of finding the derivative of the arctangent function, exploring the underlying mathematics and providing step-by-step guidance.

Understanding the Arctangent Function

The arctangent function, tan⁻¹(x), is the inverse of the tangent function. It takes a real number as input and returns an angle whose tangent is that number. In mathematical notation, this relationship can be expressed as:

tan⁻¹(x) = y ⟺ tan(y) = x

The arctangent function maps its domain from all real numbers to the interval (-π/2, π/2). This limited domain gives it a unique property and simplifies its derivative calculation.

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The Derivative of the Arctangent Function

To find the derivative of the arctangent function, we can utilize the chain rule, a fundamental rule of differentiation. The chain rule states that the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function. In the case of the arctangent function, we can consider it as the composition of two functions: the tangent function and the identity function.

The derivative of the tangent function, denoted as d/dx (tan(x)), is given by:

d/dx (tan(x)) = sec²(x)

Now, let’s find the derivative of the identity function, y = x, which is simply:

d/dx (x) = 1

Using the chain rule, we can find the derivative of the arctangent function, tan⁻¹(x):

d/dx (tan⁻¹(x)) = d/dx (tan⁻¹(u)) * d/du (u) (where u = tan(u))

Since d/du (u) = 1 and d/dx (tan⁻¹(u)) = 1 / (1 + u²) (by using the derivative of arctan), we have:

d/dx (tan⁻¹(x)) = 1 / (1 + u²)

Replacing u with tan(x), we get:

d/dx (tan⁻¹(x)) = 1 / (1 + tan²(x))

However, we know that tan²(x) + 1 = sec²(x) (by the Pythagorean identity for tangent), so:

d/dx (tan⁻¹(x)) = 1 / sec²(x) = cos²(x)

Conclusion

Understanding the derivative of the arctangent function is crucial for a deeper comprehension of calculus and its applications. By utilizing the chain rule and the properties of trigonometric functions, we’ve arrived at the result that the derivative of the arctangent function, tan⁻¹(x), is equal to cos²(x). This result has significance in fields ranging from physics and engineering to computer science and economics, wherever the concept of change is encountered. Mastery of these mathematical techniques empowers us to analyze complex functions and uncover insights that drive innovation and understanding in a wide array of disciplines.

FAQs

Q: Is the derivative of tan⁻¹(x) limited to a specific range?

A: Yes, the derivative of tan⁻¹(x) lies in the range of -π/2 to π/2 radians.

Q: Can you explain the chain rule in simple terms?

A: Certainly! The chain rule helps us differentiate composite functions. To do this, we multiply the derivative of the outer function by the derivative of the inner function.