Home
Class 12
MATHS
If log(x^2+y^2)=2tan^(-1)(y/x), show tha...

If `log(x^2+y^2)=2tan^(-1)(y/x),` show that `(dy)/(dx)=(x+y)/(x-y)` .

Text Solution

Verified by Experts

Given,`log(x^2+y^2)=2tan^(-1)(y/x)`
`d/dx(log(x^2+y^2))=d/dx(2tan^(-1)(y/x))`
=`1/(x^2+y^2)​×d/dx​(x^2+y^2)=2×1/(1+(y/x)^2)​×(d/dx)​(y/x​)`
=`1/(x^2+y^2).(2x+2y(dy/dx))=2/(x^2+y^2)(x(dy/dx)-y)`
=`(2x+2y(dy/dx))=2(x(dy/dx)-y)`
=`x+y(dy/dx)​=x(dy/dx)​−y `
...
Promotional Banner

Topper's Solved these Questions

  • DIFFERENTIALS, ERRORS AND APPROXIMATIONS

    RD SHARMA|Exercise Solved Examples And Exercises|102 Videos

  • DIRECTION COSINES AND DIRECTION RATIOS

    RD SHARMA|Exercise Solved Examples And Exercises|67 Videos

Similar Questions

Explore conceptually related problems

If log (x^(2)+y^(2))=tan^(-1)((y)/(x)), then show that (dy)/(dx)=(x+y)/(x-y)

If log sqrt(x^(2)+y^(2))=tan^(-1)((y)/(x)),then(dy)/(dx) is

If log (x^(2)+y^(2)) = 2 tan ^(-1) (x/y) " then show that " (dy)/(dx) = (y-x)/(y +x)

If x^(y)=e^(x-y), then show that (dy)/(dx)=(log x)/((1+log x)^(2))

If x^(y)=e^(x-y), show that (dy)/(dx)=(log x)/({log(xe)}^(2))

If y=tan^(-1)x, show that (1+x^(2))(d^(2)y)/(dx^(2))+2x(dy)/(dx)=0

If log y=tan^(-1)x, show that (1+x^(2))y_(2)+(2x-1)y_(1)=0

Q.if log y=sin^(-1)x, show that,(1-x^(2))((d^(2)y)/(dx^(2)))=x((dy)/(dx))+y

If log(x+y)=log(xy)+a,"show that"(dy)/(dx)=-(y^(2))/(x^(2))

If y^(x)=e^(y-x), prove that (dy)/(dx)=((1+log y)^(2))/(log y)

RD SHARMA-DIFFERENTIATION-Solved Examples And Exercises
  1. If the derivative of tan^(-1)(a+b x) takes the value 1 at x=0, prove t...

    Text Solution

    |

  2. If y=sin[2tan^(-1){sqrt((1-x)/(1+x))}],find(dy)/(dx)

    Text Solution

    |

  3. If log(x^2+y^2)=2tan^(-1)(y/x), show that (dy)/(dx)=(x+y)/(x-y) .

    Text Solution

    |

  4. If y=cos^(-1)(2x)+2cos^(-1)sqrt(1-4x^2),-1/2<x<0 , find (dy)/(dx)

    Text Solution

    |

  5. If sqrt(1-x^6)+sqrt(1-y^6)=a(x^3-y^3),p rov et h a t(dy)/(dx)=(x^2)/(...

    Text Solution

    |

  6. If cos^(-1)((x^2-y^2)/(x^2+y^2))=tan^(-1)a , prove that (dy)/(dx)=y/xd...

    Text Solution

    |

  7. Differentiate cos^(-1)(1-2x^2) with respect to x , if 0<x<1 (ii) -1<x...

    Text Solution

    |

  8. Differentiate tan^(-1)((3x-x^3)/(1-3x^2)),if -1/sqrt(3)ltxlt1 /sqrt...

    Text Solution

    |

  9. Differentiate each of the following functions with respect to x : (i)...

    Text Solution

    |

  10. Differentiate the following function with respect to x : sin^(-1)(si...

    Text Solution

    |

  11. Differentiate sin^(-1)(2xsqrt(1-x^2))w i t hr e s p e c ttox ,if -1/(...

    Text Solution

    |

  12. Differentiate sin^(-1)(3x-4x^3)

    Text Solution

    |

  13. Differentiate cos^(-1)(2x^2-1) with respect to x , if 0<x<1 (ii) -1<x...

    Text Solution

    |

  14. If y=cos^(-1){xsqrt(1-x)+sqrt(x)sqrt(1-x^2)} and 0<x<1 find (dy)/(dx)

    Text Solution

    |

  15. If y=sqrt(a^2-x^2) , prove that y(dy)/(dx)+x=0

    Text Solution

    |

  16. If y=e^x+e^(-x),"p r o v et h a t"(dy)/(dx)=sqrt(y^2-4)

    Text Solution

    |

  17. If y=sqrt(x^2+a^2) ,prove that y(dy)/(dx)-x=0

    Text Solution

    |

  18. If y=xsin^(-1)x+sqrt(1-x^2), p rov et h a t(dy)/(dx)=sin^(-1)x

    Text Solution

    |

  19. If y=1/2log((1-cos2x)/(1+cos2x)),p rov et h a t(dy)/(dx)=2cos e c2x

    Text Solution

    |

  20. If y=e^xcosx ,p rov et h a t(dy)/(dx)=sqrt(2)e^xcos(x+pi/4)

    Text Solution

    |