InterviewSolution
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InterviewSolution
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If `y=sin^(-1)((4x)/(1+4x^(2)))`, then what is `(dy)/(dx)` equal to?A. `(1)/(1+4x^(2))`B. `-(1)/(1+4x^(2))`C. `(4)/(1+4x^(2))`D. `(4x)/(1+4x^(2))` |
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Answer» Correct Answer - C Let` y=sin^(-1)((4x)/(1+4x^(2)))=sin^(-1)((2.2x)/(1+(2x)^(2)))` Put `2x=tan theta rArr theta =tan^(-1)2x` `therefore y=sin^(-1)((2tan theta)/(1+tan^(2)theta))` `=sin^(-1)(sin2 theta)=2 theta(because sin 2 theta=(2tan theta)/(1+tan^(2)theta))` `=2tan^(-1)2x` On differentiating w.r.t. x, we get `(dy)/(dx)=(2)/(1+(2x)^(2)).2=(4)/(1+4x^(2))` ALTERNATESOLUTION `y=sin^(-1)((4x)/(1+4x^(2)))` `(dy)/(dx)=(1)/(sqrt(1-((4x)/(1+4x^(2)))^(2)))xx((1+4x^(2))4-4x(8x))/((1+4x^(2))^(2))` `=(4-16x^(2))/((1+4x^(2))sqrt(1-8x^(2)+16x^(4)))` `=(4-16x^(2))/((1+4x^(2))(1-4x^(2)))` `=((2)^(2)-(4x)^(2))/((1+4x^(2))(1-2x)(1+2x))=((2+4x)2)/((1+4x^(2))(1+2x))=(4)/(1+4x^(2))` |
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