Question

Differentiate the given functions.

1. g(x)=(1)/(2)sec^(-1)(4)/(x)+2csc^(-1)4x
2. (x)=6sin^(-1)\sqrt(1-x^(2))
3. h(x)=xsin^(-1)2x
4. y=sec^(-1)\sqrt(x^(2))+4
5. y=cos^(-1)(sinx)

Answer 1

$(1)\\g(x)=\dfrac{1}{sec^{-1}(\frac{4}{x})+csc^{-1}(4x)}\\ \text{Differentiating wrt x on both sides,}\\ g'(x)\\=-\dfrac{1}{(sec^{-1}(\frac{4}{x})+csc^{-1}(4x))^2}\times \dfrac{d}{dx}(sec^{-1}(\frac{4}{x})+csc^{-1}(4x))\\ =-\dfrac{1}{(sec^{-1}(\frac{4}{x})+csc^{-1}(4x))^2}\times \left(\dfrac{x^2}{4\sqrt{16-x^2}}\left(-\dfrac{4}{x^2}\right)-\dfrac{1}{4|x|\sqrt{16x^2-1}{}}\right)\\\;\\ = \boxed{\dfrac{1}{(sec^{-1}(\frac{4}{x})+csc^{-1}(4x))^2}\times \left(\dfrac{1}{\sqrt{16-x^2}}+\dfrac{1}{4|x|\sqrt{16x^2-1}{}}\right)}\\ \;\\ (2)$

$f(x)=6sin^{-1}\sqrt{1-x^2}\\ \text{Differentiating wrt x on both sides,}\\ f'(x)\\ =\dfrac{6}{\sqrt{1-\left(\sqrt{1-x^2}\right)^2}}\times\dfrac{d}{dx}\left(\sqrt{1-x^2}\right)\\ =\dfrac{6}{|x|}\times \dfrac{-2x}{2\sqrt{1-x^2}}\\\;\\ =\boxed{-\dfrac{6x}{|x|\sqrt{1-x^2}}}\\\;\\ (3)$

$h(x)=xsin^{-1}(2x)\\ \text{Differentiating wrt x on both sides,}\\ h'(x)\\ =\dfrac{x}{\sqrt{1-4x^2}}\times2+sin^{-1}(2x)\\\;\\ =\boxed{\dfrac{2x}{\sqrt{1-4x^2}}+sin^{-1}(2x)} \\\;\\ (4)$

$y=sec^{-1}\sqrt{x^2}\\ \text{Differentiating wrt x on both sides,}\\ \dfrac{dy}{dx}\\ =\dfrac{1}{|x|\sqrt{x^2-1}}\times \dfrac{2x}{2\sqrt{x^2}}\\\;\\ =\dfrac{x}{x^2\sqrt{x^2-1}}=\boxed{\dfrac{1}{x\sqrt{x^2-1}}}$

$(5)\\ y=cos^{-1}(sinx)\\ \text{Differentiating wrt x on both sides,} \\ \dfrac{dy}{dx}\\ = -\dfrac{1}{\sqrt{1-sin^2x}}\times cosx\\=\boxed{-\dfrac{cosx}{|cosx|}}$