Derivative of arctan(x): Application of the Chain Rule and Simplification

d/dx arctan(x)

To find the derivative of the function f(x) = arctan(x), we can use the chain rule

To find the derivative of the function f(x) = arctan(x), we can use the chain rule. Let’s start by writing the derivative of arctan(x) as d/dx(arctan(x)).

The chain rule states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

In this case, f(u) = arctan(u) and g(x) = x.

Applying the chain rule, we have:

d/dx(arctan(x)) = f'(g(x)) * g'(x)

Now, we need to find the derivative of arctan(u) with respect to u, which is denoted as f'(u). The derivative of arctan(u) is given by:

f'(u) = 1 / (1 + u^2)

Since g(x) = x, the derivative of g(x) with respect to x, which is denoted as g'(x), is simply 1.

Now, let’s substitute these values into the equation:

d/dx(arctan(x)) = f'(g(x)) * g'(x)

d/dx(arctan(x)) = 1 / (1 + g(x)^2) * g'(x)

d/dx(arctan(x)) = 1 / (1 + x^2) * 1

Therefore, the derivative of arctan(x), or d/dx(arctan(x)), is equal to 1 / (1 + x^2).

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

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