Master the Differentiation of Arctan(x) with the Chain Rule: Derivative of Arctan(x) Unveiled

d/dx arctan(x)

To differentiate the function arctan(x) with respect to x, we will use the chain rule

To differentiate the function arctan(x) with respect to x, we will use the chain rule. Recall that the chain rule states that if we have a function composition f(g(x)), then the derivative is given by (f'(g(x))) * g'(x).

Now, let’s apply the chain rule to differentiate arctan(x).

We start by letting u = arctan(x). This means that u is the inverse function of tan(x), and x = tan(u).

To find du/dx, we need to take the derivative of both sides of x = tan(u) with respect to x.

Using implicit differentiation, we have:

1 = sec^2(u) * du/dx

Now, we can solve for du/dx:

du/dx = 1 / sec^2(u)

Since sec^2(u) is equal to 1 + tan^2(u), we can rewrite du/dx as:

du/dx = 1 / (1 + tan^2(u))

And because x = tan(u), we can substitute x for tan(u) in the previous expression to get:

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

So, the derivative of arctan(x) with respect to x is:

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

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

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