Answer to Question #153669 in Calculus for che

Question #153669

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)
1
Expert's answer
2021-01-18T19:27:51-0500

(1)g(x)=1sec1(4x)+csc1(4x)Differentiating wrt x on both sides,g(x)=1(sec1(4x)+csc1(4x))2×ddx(sec1(4x)+csc1(4x))=1(sec1(4x)+csc1(4x))2×(x2416x2(4x2)14x16x21)  =1(sec1(4x)+csc1(4x))2×(116x2+14x16x21)  (2)(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)=6sin11x2Differentiating wrt x on both sides,f(x)=61(1x2)2×ddx(1x2)=6x×2x21x2  =6xx1x2  (3)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)=xsin1(2x)Differentiating wrt x on both sides,h(x)=x14x2×2+sin1(2x)  =2x14x2+sin1(2x)  (4)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=sec1x2Differentiating wrt x on both sides,dydx=1xx21×2x2x2  =xx2x21=1xx21y=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=cos1(sinx)Differentiating wrt x on both sides,dydx=11sin2x×cosx=cosxcosx(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|}}


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