Calculating the Derivative of Arctan(x) Using the Chain Rule and Trigonometric Identities

Derivative of arctan(x)

The derivative of the arctan(x) function can be found using the chain rule of differentiation

The derivative of the arctan(x) function can be found using the chain rule of differentiation. The arctan(x) function represents the inverse tangent function, which gives the angle whose tangent is x.

To find the derivative of arctan(x), denoted as d/dx (arctan(x)), we proceed as follows:

Let y = arctan(x).

Taking the tangent of both sides, we have: tan(y) = x.

Now differentiate both sides with respect to x using the chain rule:

sec^2(y) * dy/dx = 1.

Rearranging for dy/dx, we get:

dy/dx = 1/sec^2(y).

To find an expression for dy/dx in terms of x, we need to relate the sec^2(y) to x using the given equation tan(y) = x.

By the trigonometric identity, sec^2(y) = 1 + tan^2(y), we can rewrite the equation as:

dy/dx = 1/(1 + tan^2(y)).

Substituting the value of tan(y) = x, we have:

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

Therefore, the derivative of arctan(x) is 1/(1 + x^2).

This derivative represents the rate of change of the angle whose tangent is x, with respect to x.

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