Derivative of Tan Inverse x – Definition With Examples

Welcome to Brighterly, your go-to hub for transforming the intimidating universe of mathematics into a playground of fun, exploration, and learning. Today, we invite our budding mathematicians to delve into the intriguing world of calculus, as we illuminate the concept of the derivative of tan inverse x, or arctan(x). From breaking down the definition to providing relatable examples, we’ll make the journey as engaging and enlightening as possible!

What is the Derivative of Tan Inverse x?

In the mysterious realm of calculus, the derivative of tan inverse x is a key concept, rooted in the parent function, tan inverse x, or arctan(x). Simply put, the derivative quantifies how a function changes as its input changes. The derivative of tan inverse x is 1/(1 + x²). This equation gives us the slope of the tangent line to the function at any point (x, arctan(x)).

Definition of Tan Inverse x

To fully appreciate the concept, let’s first understand Tan Inverse x or arctan(x). It is the inverse function of the tangent function. If y = tan(x), then x = arctan(y). In other words, it is the angle whose tangent is x. This function is critical in trigonometry, helping solve triangles and analyse oscillations.

Definition of Derivative

Now, let’s understand derivatives. They measure how a function changes as its inputs change. Like a speedometer that measures the speed of a car, a derivative provides the ‘rate of change’ or the ‘slope’ of a function at any given point. In this context, the derivative of tan inverse x calculates how quickly the function arctan(x) changes as x changes.

Understanding the Derivative of Tan Inverse x

The derivative of tan inverse x is a result of the use of the Chain Rule, a fundamental theorem in calculus. It represents how steep the curve is at any given point on the graph of y = arctan(x). When x increases or decreases, how much does arctan(x) increase or decrease? The derivative gives you that rate of change.

Properties of Tan Inverse x

Like any mathematical function, tan inverse x has its unique properties. For instance, it is an odd function, meaning arctan(-x) = -arctan(x). Also, its range is from -π/2 to π/2, and its graph is always increasing, showcasing how an understanding of these properties leads to more effective problem-solving.

Properties of Derivatives

Derivatives also come with their set of properties that simplify complex problems. They are linear operators, meaning the derivative of a sum of functions is the sum of their derivatives. Also, the derivative of a constant times a function is equal to the constant times the derivative of the function.

Differences between Tan Inverse x and Its Derivative

Although closely related, there are key differences between tan inverse x and its derivative. Tan inverse x is the original function, representing an angle whose tangent is x. For instance, if we consider an angle y whose tangent is 1, using tan inverse x, we can say y = arctan(1) = π/4 or 45°.

The derivative, on the other hand, is a measure of how this function changes with x. Consider a small change in x (Δx). The corresponding change in y (Δy) will be given by dy/dx * Δx. For instance, if we take x = 1 (from the previous example), the derivative at this point is 1/(1 + 1²) = 0.5. So, if x changes by a small amount (say 0.01), the change in y will be approximately 0.5 * 0.01 = 0.005.

This distinction means that while arctan(x) varies between -π/2 and π/2, its derivative is defined for all real numbers.

Equation of the Derivative of Tan Inverse x

In calculus, the equation of the derivative of tan inverse x is given by 1/(1 + x²). This equation was derived using the Chain Rule, which lets us differentiate a wide variety of complex functions.

For example, if we have y = arctan(2x), the derivative would be dy/dx = (2)/(1 + (2x)²), showing how the derivative of a composite function involves both the derivative of the outer function and the inner function.

Writing the Equation of Tan Inverse x

Writing the equation of tan inverse x is fairly straightforward. If y is the angle whose tangent is x, the equation is written as y = arctan(x) or y = tan⁻¹(x).

For example, consider a right-angled triangle where the opposite side is 3 units and the adjacent side is 4 units. The tangent of the angle (y) opposite the side of length 3 would be 3/4 = 0.75. So, we can say y = arctan(0.75).

Writing the Equation for the Derivative of Tan Inverse x

Writing the equation for the derivative is equally simple. Given y = arctan(x), the derivative (dy/dx) is given by 1/(1 + x²).

For example, if we have y = arctan(3x), the derivative would be dy/dx = (3)/(1 + (3x)²), again showing the application of the Chain Rule in finding the derivative of the tan inverse function.

Practice Problems on the Derivative of Tan Inverse x

Let’s bring theory to practice with some problems.

  1. Problem: Calculate the derivative of arctan(3x). Solution: The derivative is dy/dx = (3)/(1 + (3x)²).

  2. Problem: Calculate the derivative of arctan(sqrt(x)). Solution: This is a little trickier. Here, we use the chain rule again. If we let u = sqrt(x), then we have y = arctan(u). The derivative of y with respect to u is 1/(1 + u²), and the derivative of u with respect to x is 1/(2sqrt(x)). So, the overall derivative is dy/dx = (1/(2sqrt(x)))/(1 + x).

Conclusion

That’s a wrap on our enlightening journey exploring the derivative of tan inverse x! We hope the voyage was as stimulating for you as it was for us here at Brighterly. Remember, in the world of mathematics, every formula, every equation, every concept has a story to tell. The derivative of tan inverse x is not just a random assortment of symbols—it is a testament to the power of human curiosity, a gateway to an infinite cosmos of learning.

As you continue your exploration, remember to enjoy the process. Maths, at its core, is not about rote memorization—it’s about understanding, curiosity, and the joy of discovery. And always remember, whether it’s grasping the properties of derivatives or tackling tan inverse x, Brighterly is here to light your path, turning complex concepts into comprehensible knowledge, and making learning a delightful journey rather than a daunting task.

Frequently Asked Questions on the Derivative of Tan Inverse x

We believe that questions are the stepping stones to knowledge. So, here are the answers to some Frequently Asked Questions that might help clear up any lingering confusion.

What is the derivative of tan inverse 2x?

The derivative of tan inverse 2x, following the chain rule, is 2/(1 + (2x)²). Here, the outer function is arctan(x), and the inner function is 2x. The derivative is the derivative of the outer function, evaluated at the inner function, multiplied by the derivative of the inner function.

How to use the Chain Rule in the derivative of tan inverse x?

The Chain Rule is a fundamental tool in calculus, allowing us to differentiate a vast variety of functions. To apply the Chain Rule to the derivative of tan inverse x, we first identify the inner and outer functions. If we have a composite function like arctan(f(x)), the outer function is arctan(x) and the inner function is f(x). The chain rule states that the derivative of this composite function is the derivative of the outer function evaluated at the inner function, times the derivative of the inner function. So, the derivative of arctan(f(x)) is f'(x)/(1 + (f(x))²).

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