Question

Differentiate $y=\left[{\sin }^{-1}(x-\frac{4x^3}{27})\right]$ w.r.t $x$

Answer 1

$y=\left[{\sin }^{-1}(x-\frac{4x^3}{27})\right]$

$\Rightarrow \sin y=x-\frac{4x^3}{27}$

$\Rightarrow \cos y\frac{dy}{dx}=1-\frac{4}{27}\left(3x^2\right)$

$\Rightarrow \cos y\frac{dy}{dx}=1-\frac{4x^2}{9}$

$\Rightarrow \frac{dy}{dx}=\frac{1-\frac{4x^2}{9}}{\cos y}$

$\Rightarrow \frac{dy}{dx}=\frac{1-\frac{4x^2}{9}}{\sqrt{1-{\sin }^{2}y}}$

$\Rightarrow \frac{dy}{dx}=\frac{1-\frac{4x^2}{9}}{\sqrt{1-{(x-\frac{4x^3}{27})}^2}}$

$\Rightarrow \frac{dy}{dx}=\frac{27(9-4x^2)}{9\sqrt{({27}^2-{(27x-4x^3)}^2)}}$

$\Rightarrow \frac{dy}{dx}=\frac{3(9-4x^2)}{\sqrt{729-729x^2-16x^6+216x^4}}$

$\therefore \frac{dy}{dx}=\frac{3(9-4x^2)}{\sqrt{729-729x^2+216x^4-16x^6}}$