Step-by-Step Calculation | Finding the Derivative of the Inverse Tangent Function

derivative of tan^-1

The derivative of the inverse tangent function, also written as “tan^(-1)” or “arctan,” can be found using calculus techniques

The derivative of the inverse tangent function, also written as “tan^(-1)” or “arctan,” can be found using calculus techniques. Let’s calculate it step by step.

We start by recalling the derivative of the inverse function. If we have a function y = f(x) and its inverse function is x = f^(-1)(y), then the derivative of the inverse function is given by:

dy/dy = 1 / (dx/dy)

Applying this concept to the inverse tangent function, we have:

Let y = tan^(-1)(x)

To find dy/dx (derivative of y with respect to x), we’ll differentiate both sides of the above equation.

d/dx(y) = d/dx(tan^(-1)(x))

Now, we need to apply the chain rule. The chain rule allows us to differentiate compositions of functions.

The derivative of the inverse tangent function can be expressed as:

dy/dx = (d/dx(tan^(-1)(x))) / (dx/dx)

Since dx/dx is simply 1, the chain rule simplifies to:

dy/dx = d/dx(tan^(-1)(x))

To differentiate the inverse tangent function, we can use the fact that:

d/dx(tan^(-1)(x)) = 1 / (1 + x^2)

Therefore, the derivative of tan^(-1)(x) is:

dy/dx = 1 / (1 + x^2)

And that’s the final answer! The derivative of the inverse tangent function is 1 / (1 + x^2).

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