Differentiate ^ -1 (8 x 1-15 x^2 ) w.r. to x - Vidyadip

Question

Differentiate $\tan ^{-1}\left(\frac{8 x}{1-15 x^2}\right)$ w.r. to $x$

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Answer

$ \text { Let } y =\tan ^{-1}\left(\frac{8 x}{1-15 x^2}\right)$
$=\tan ^{-1}\left(\frac{5 x+3 x}{1-(5 x)(3 x)}\right)$
$=\tan ^{-1} 5 x +\tan ^{-1} 3 x $
Differentiating w. r. t. x, we get
$ \frac{ d y}{ d x}=\frac{ d }{ d x}\left(\tan ^{-1} 5 x+\tan ^{-1} 3 x\right)$
$=\frac{1}{1+(5 x)^2} \cdot \frac{ d }{ d x}(5 x)+\frac{1}{1+(3 x)^2} \cdot \frac{ d }{ d x}(3 x)$
$=\frac{1}{1+25 x^2} \cdot(5)+\frac{1}{1+9 x^2} \cdot 3$
$\therefore \frac{ d y}{ d x}=\frac{5}{1+25 x^2}+\frac{3}{1+9 x^2} $

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