derivative of tan^-1
1/(1 + x^2)
The derivative of arctan (tan^-1) is:
d/dx arctan(x) = 1 / (1 + x^2)
This can be derived using the formula for the derivative of the inverse function:
d/dx f^-1(x) = 1 / f'(f^-1(x))
Where f(x) = tan(x), and f'(x) = sec^2(x)
So, substituting into the formula, we get:
d/dx arctan(x) = 1 / sec^2(arctan(x))
Using the Pythagorean identity, we can simplify:
sec^2(arctan(x)) = 1 + tan^2(arctan(x)) = 1 + x^2
Substituting this back into the previous equation, we get:
d/dx arctan(x) = 1 / (1 + x^2)
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