Derivative of arctan(x) | Calculating the Derivative of tan^(-1)(x) using the Chain Rule

Derivative of tan^-1

To find the derivative of the function f(x) = tan^(-1)(x), or arctan(x), we can use the chain rule

To find the derivative of the function f(x) = tan^(-1)(x), or arctan(x), we can use the chain rule.

Let’s start by writing the function in terms of the inverse trigonometric function:

f(x) = arctan(x)

Now, we can rewrite the function using the trigonometric identity:

tan(f(x)) = x

Now, let’s take the derivative of both sides with respect to x.

d/dx[tan(f(x))] = d/dx[x]

Using the chain rule, we have:

sec^2(f(x)) * d/dx[f(x)] = 1

We can simplify this equation by using the identity:

sec^2(f(x)) = 1 + tan^2(f(x))

Substituting this back in, we have:

(1 + tan^2(f(x))) * d/dx[f(x)] = 1

Now, let’s solve for d/dx[f(x)], which is the derivative of arctan(x):

d/dx[f(x)] = 1 / (1 + tan^2(f(x)))

We can rewrite tan^2(f(x)) as x^2, since tan(f(x)) = x:

d/dx[f(x)] = 1 / (1 + x^2)

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

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

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