The Derivative of Arctan(x) with Respect to x | Step-by-Step Calculation

d/dx arctan(x)

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

To find the derivative of arctan(x), we can use the chain rule. Let’s go step by step:

First, recall the definition of the arctan function:

arctan(x) = tan^(-1)(x)

Now, we can differentiate both sides of this equation with respect to x:

d/dx(arctan(x)) = d/dx(tan^(-1)(x))

Using the chain rule, we can express this derivative in terms of the derivative of the arctan function and the derivative of the inner function, x.

Let u(x) = tan^(-1)(x) and v(x) = x.

The chain rule states that if y = u(v(x)), then dy/dx = du/dv * dv/dx.

Applying this to our situation:

d/dx(arctan(x)) = (du/dv) * (dv/dx)

Now, let’s differentiate each part:

d/dx(arctan(x)) = (d(u)/dv) * (dv/dx)

The derivative of u with respect to v is computed by differentiating the arctan function, which is 1/(1 + v^2). Replacing v with x, we get:

d(u)/dv = 1/(1 + x^2)

The derivative of v with respect to x is simply 1.

So, our equation becomes:

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

Simplifying further, the final result is:

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

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

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