find nth derivative of tan^{-1}x

To find the nth derivative of tan^{-1}x, we can use the power rule for derivatives.

Step 1: Start by finding the first derivative of tan^{-1}x.

The derivative of tan^{-1}x can be found using the chain rule. Let's denote y = tan^{-1}x.

dy/dx = d(tan^{-1}x)/dx

Using the chain rule, we have:

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

Simplifying, we get:

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

Step 2: Now, we need to find the second derivative.

To find the second derivative, we differentiate the first derivative with respect to x.

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

Using the quotient rule, we have:

d^2y/dx^2 = (-2x)/(1+x^2)^2

Step 3: Continuing this process, we can find the nth derivative.

To find the nth derivative, we differentiate the (n-1)th derivative with respect to x.

d^n y/dx^n = d(d^(n-1)y/dx^(n-1))/dx

Using the quotient rule and simplifying, we get:

d^n y/dx^n = (-1)^(n-1) * (n-1)! * (2^(n-1)) * x^(n-1) / (1+x^2)^n

Therefore, the nth derivative of tan^{-1}x is given by:

d^n y/dx^n = (-1)^(n-1) * (n-1)! * (2^(n-1)) * x^(n-1) / (1+x^2)^n