Determine the second derivative of y=arctanx, if x is in the positive real set.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to find the second derivative of arc tan (x)

[arc tan (x)]'' = [1/(1 + x^2)]'

[1/(1 + x^2)]' = [(1 + x^2)^-1]'

=> -1*(1 + x^2)^-2*2x

=> -2x/(1 + x^2)^2

The second integral of arc tan x =  -2x/(1 + x^2)^2

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

For the beginning, we'll calculate the first derivative of f(x):

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

Now, we'll calculate f"(x) of the expression arctan x, or we'll calculate the first derivative of f'(x).

f"(x) = [f'(x)]'

Since f'(x) is a ratio, we'll apply the quotient rule:

f"(x) = [1'*(1+x^2)-1*(1+x^2)']/(1+x^2)^2

We'll put 1' = 0 and we'll remove the brackets:

f"(x)= -2x/(1+x^2)^2

Because of the fact that denominator is always positive, then the numerator will influence the ratio.

We notice that because of the fact that numerator is negative over the interval [0,infinite) => f"(x)<0.

Therefore, the 2nd derivative of the function y=arctan x is y"= -2x/(1+x^2)^2 and the graph of the function is concave over the positive real set of numbers.

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