What is the derivative of tan-1(x)?
Understand the Problem
The question is asking for the derivative of the inverse tangent function, denoted as tan^(-1)(x) or arctan(x). The derivative of this function is a common calculus problem, where we will use the formula for the derivative of arctan(x).
Answer
The derivative of the inverse tangent function is $$ \frac{dy}{dx} = \frac{1}{1 + x^2} $$
Answer for screen readers
The derivative of the inverse tangent function is $$ \frac{dy}{dx} = \frac{1}{1 + x^2} $$
Steps to Solve
- Identify the Function and Formula
We start with the function we need to differentiate, which is $y = \tan^{-1}(x)$ or $y = \arctan(x)$.
- Use the Derivative Formula
The standard formula for the derivative of arctan(x) is: $$ \frac{dy}{dx} = \frac{1}{1 + x^2} $$
- Compute the Derivative
Now we apply the derivative formula directly: For $y = \tan^{-1}(x)$, we substitute $x$ into the formula: $$ \frac{dy}{dx} = \frac{1}{1 + x^2} $$
The derivative of the inverse tangent function is $$ \frac{dy}{dx} = \frac{1}{1 + x^2} $$
More Information
The derivative of arctan(x) is widely used in calculus, especially in integration and solving trigonometric equations. It also plays an important role in various applications of mathematics and engineering.
Tips
- Confusing the derivative of arctan(x) with that of the regular tangent function; remember they are different.
- Forgetting to simplify or rewrite expressions appropriately after finding derivatives.