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)=\\dfrac{1}{sec^{-1}(\\frac{4}{x})+csc^{-1}(4x)}\\\\\n\\text{Differentiating wrt x on both sides,}\\\\\ng'(x)\\\\=-\\dfrac{1}{(sec^{-1}(\\frac{4}{x})+csc^{-1}(4x))^2}\\times \\dfrac{d}{dx}(sec^{-1}(\\frac{4}{x})+csc^{-1}(4x))\\\\\n=-\\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)\\\\\\;\\\\\n=\n\\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)}\\\\\n\\;\\\\\n(2)"

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

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

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


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


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