1) if y=sqrt((x^2)+1)-log[1/x +sqrt(1+1/(x^2))],find dy/dx.
2)find the equation of tangent to the curve y=sqrt(4x-2) which is parallel to the line 4x-2y+5=0.

Asked by Adeeb Khan | 16th Jul, 2014, 01:00: AM

Expert Answer:

1. space y equals square root of x squared plus 1 end root minus log open square brackets 1 over x plus square root of 1 plus 1 over x squared end root close square brackets fraction numerator d y over denominator d x end fraction equals fraction numerator 2 x over denominator 2 square root of x squared plus 1 end root end fraction minus fraction numerator 1 over denominator open square brackets 1 over x plus square root of 1 plus 1 over x squared end root close square brackets end fraction cross times open square brackets fraction numerator minus 1 over denominator x squared end fraction plus fraction numerator begin display style fraction numerator minus 2 over denominator x cubed end fraction end style over denominator 2 square root of 1 plus 1 over x squared end root end fraction close square brackets fraction numerator d y over denominator d x end fraction equals fraction numerator x over denominator square root of x squared plus 1 end root end fraction plus open square brackets fraction numerator x over denominator 1 plus square root of x squared plus 1 end root end fraction close square brackets cross times open square brackets fraction numerator square root of x squared plus 1 end root plus 1 over denominator x squared open square brackets square root of x squared plus 1 end root close square brackets end fraction close square brackets fraction numerator d y over denominator d x end fraction equals fraction numerator x over denominator square root of x squared plus 1 end root end fraction plus fraction numerator 1 over denominator x open square brackets square root of x squared plus 1 end root close square brackets end fraction equals fraction numerator x squared plus 1 over denominator x open square brackets square root of x squared plus 1 end root close square brackets end fraction equals fraction numerator open square brackets square root of x squared plus 1 end root close square brackets over denominator x end fraction equals square root of 1 plus 1 over x squared end root

2. space y equals square root of 4 x minus 2 end root fraction numerator d y over denominator d x end fraction equals fraction numerator 4 over denominator 2 square root of 4 x minus 2 end root end fraction equals fraction numerator 2 over denominator square root of 4 x minus 2 end root end fraction  G i v e n space l i n e space 4 x minus 2 y plus 5 equals 0 comma space s l o p e equals 2 equals fraction numerator d y over denominator d x end fraction fraction numerator d y over denominator d x end fraction equals fraction numerator 2 over denominator square root of 4 x minus 2 end root end fraction equals 2  square root of 4 x minus 2 end root equals 1 4 x minus 2 equals 1 4 x equals 3 x equals 3 over 4 rightwards double arrow y equals 1  y minus 1 equals 2 open parentheses x minus 3 over 4 close parentheses 2 y minus 2 equals 4 x minus 3 4 x minus 2 y equals 1 space i s space t h e space r e q u i r e d space tan g e t space t o space t h e space g i v e n space c u r v e

Answered by Dharma Teja | 16th Jul, 2014, 09:40: AM