Differentiate-with-respect-to-x-arctan-a-2-x-2-a-2-x-2-

Question Number 45464 by peter frank last updated on 13/Oct/18

D ifferentiate with respect to x

\arctan (\frac{a^{2} + x^{2}}{a^{2} - x^{2}})

Commented by peter frank last updated on 14/Oct/18

thank sir . really appreciate

Commented by maxmathsup by imad last updated on 14/Oct/18

y o u a r e w e l c o m e s i r

Commented by maxmathsup by imad last updated on 14/Oct/18

l e t f (x) = a r c t a n (\frac{a^{2} + x^{2}}{a^{2} - x^{2}}) = a r c t a n (u (x)) \Rightarrow f^{^{'}} (x) = \frac{u^{^{'}} (x)}{1 + u^{2} (x)} b u t

u^{^{'}} (x) = \frac{2 x (a^{2} - x^{2}) - (a^{2} + x^{2}) (- 2 x)}{{(a^{2} - x^{2})}^{2}} = \frac{2 a^{2} x - 2 x^{3} + 2 a^{2} x + 2 x^{3}}{{(a^{2} - x^{2})}^{2}}

= \frac{4 a^{2} x}{{(a^{2} - x^{2})}^{2}} \Rightarrow f^{^{'}} (x) = \frac{4 a^{2} x}{{(a^{2} - x^{2})}^{2}} . \frac{1}{1 + \frac{{(a^{2} + x^{2})}^{2}}{{(a^{2} - x^{2})}^{2}}}

= \frac{4 a^{2} x}{{(a^{2} - x^{2})}^{2}} \frac{1}{{(a^{2} - x^{2})}^{2} + (a^{2} + x^{2})} {(a^{2} - x^{2})}^{2} = \frac{4 a^{2} x}{a^{4} - 2 a^{2} x^{2} + x^{4} + a^{4} + 2 a^{2} x^{2} + x^{4}}

= \frac{4 a^{2} x}{2 a^{4} + 2 x^{4}} \Rightarrow ★ f^{^{'}} (x) = \frac{2 a^{2} x}{x^{4} + a^{4}} ★ .

Answered by ajfour last updated on 13/Oct/18

l e t y = \tan^{- 1} (\frac{a^{2} + x^{2}}{a^{2} - x^{2}})

\frac{d y}{d x} = \frac{1}{1 + {(\frac{a^{2} + x^{2}}{a^{2} - x^{2}})}^{2}} \times \frac{2 x (a^{2} - x^{2}) + 2 x (a^{2} + x^{2})}{{(a^{2} - x^{2})}^{2}}

\frac{d y}{d x} = \frac{2 a^{2} x}{a^{4} + x^{4}} .

Commented by peter frank last updated on 13/Oct/18

thank you sir so much . but why you differentiate numerator only .

Commented by ajfour last updated on 13/Oct/18

i t {i s n}^{'} t s o .

Commented by peter frank last updated on 14/Oct/18

okay sir i understand now . quetient rule method used .

Commented by peter frank last updated on 14/Oct/18

sir help Qn 45514