Differentaite function w.r.t x : ` f(x) = (xsqrt(x^(2)+4))/((3x+4)^(2/3)) , x gt 0`

Explanation

` y=(xsqrt(x^(2)+4))/((3x+4)^(2/3)) `, taking logarithm on both the sides, we have
`log y = =log{(xsqrt(x^(2)+4))/((3x+4)^(2/3))} = logx = 1/2log (x^(2)+4)-2/3 log (3x +4)`
Differentiating w.r.t x, we get
`1/y (dy)/(dx) = 1/x + 1/2 1/(x^(2)+4) (2x) - 2/3 ((3(1)+0))/(3(3x+4))`
` (dy)/(dx) =y[ 1/x+ x/(x^(2)+4) - 2/((3x+4))]`
`or (dy)/(dx) = (xsqrt(x^(2)+4))/((3x+4)^(2/3)) { 1/x + x/(x^(2)+4) - 2/(3x+4)}`