Question

Differentiation of tan inverse x is ?

Answer 1

we have to find differentiation of $tan^{-1}x$

Let $f(x)=tan^{-1}x$
we know, from inverse trigonometric functions,
$tan^{-1}(tanx)=x$

put x = tanx
then, $f(tanx)=tan^{-1}(tanx)=x$
f(tanx) = x
now differentiate both sides with respect to x.
f'(tanx)sec²x = 1
f'(tanx) = 1/sec²x

but we know, sec²A = 1 + tan²A from trigonometric identities
so, sec²x = 1 + tan²x

so, f'(tanx) = 1/(1 + tan²x)

now, put tanx = x

then, f'(x) = 1/(1 + x²)

hence, differentiation of tan^-1x is 1/(1 + x²)

Answer 2

Answer:

The derivative of tan inverse of x is

1 / (1+x^2)

In the attachment I have answered this problem.

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