Differentiate the following function with respect to `x :` `tan^(-1)(x/(1+6x^2))`

Explanation

Let, `y=tan ^{-1}(frac{x}{1+6 x^{2}})`

`Rightarrow y=tan ^{-1}(frac{3 x-2 x}{1+(3 x)(2 x)})`

`Rightarrow y=tan ^{-1} 3 x-tan ^{-1} 2 x[.` Since, `.tan ^{-1} x-tan ^{-1} y=tan ^{-1}(frac{x-y}{1+x y})]`

Differentiate it with respect to `x` using chain rule;

`frac{d y}{d x}=frac{1}{1+(3 x)^{2}} frac{d}{d x}(3 x)-frac{1}{1+(2 x)^{2}} frac{d}{d x}(2 x)`

`Rightarrow frac{d y}{d x}=frac{1}{1+9 x^{2}}(3)-frac{1}{1+4 x^{2}}(2)`

`therefore frac{d y}{d x}=frac{3}{1+9 x^{2}}-frac{2}{1+4 x^{2}}`