Mastering The Derivative Of Inverse Tangent Function (Tan⁻¹) In Math: Formula And Applications

derivative of tan^-1

1/(1 + x^2)

The derivative of the inverse tangent function (tan⁻¹) can be calculated using the formula:

d/dx tan⁻¹(x) = 1 / (1 + x²)

This formula says that the rate of change of the inverse tangent function with respect to its input is equal to 1 divided by 1 plus the input squared (x²).

This derivative allows us to find the slope of the tangent line to the inverse tangent function at any given point. For instance, if we want to find the slope of the tangent line to tan⁻¹ at x = 2, we can plug in x = 2 into the formula:

d/dx tan⁻¹(x) = 1 / (1 + x²)

d/dx tan⁻¹(2) = 1 / (1 + 2²) = 1/5

So the slope of the tangent line to tan⁻¹ at x = 2 is 1/5.

Note that the derivative of tan⁻¹(x) is also sometimes written as:

d/dx tan⁻¹(x) = 1 / (x² + 1)

The two formulas are equivalent and can be used interchangeably depending on the context.

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