InterviewSolution
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InterviewSolution
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The derivative of `tan^(-1)((sqrt(1+x^2)-1)/x)`with respect to `tan^(-1)((2xsqrt(1-x^2))/(1-2x^2))`at `x=0`is`1/8`(b) `1/4`(c) `1/2`(d) 1A. `1//8`B. `1//4`C. `1//2`D. 1 |
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Answer» `"Let "y=tan^(-1)((sqrt(1+x^(2))+1)/(x)) and z = tan^(-1)((2xsqrt(1-x^(2)))/(1-2x^(2)))` Putting x = tan `theta` in y, we get `y=tan^(-1)((sectheta-1)/(tan theta))=tan^(-1)(tan""(theta)/(2))=(1)/(2)tan^(-1)x` `therefore" "(dy)/(dx)=(1)/(2(1+x^(2)))` Putting `x= sin theta` in z, we get `z=tan^(-1)""((2 sin theta cos theta)/(cos 2theta))=tan^(-1)(tan 2theta)=2theta=2sin^(-1)x` `therefore" "(dz)/(dx)=(2)/(sqrt(1-x^(2)))` `"Thus, "(dy)/(dx)=((dy)/(dx))/((dz)/(dx))=(1)/(4(1+x^(2)))sqrt(1-x^(2))or ((dy)/(dz))_(x=0)=(1)/(4)` |
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