What is the derivative of #y = (xsqrt(x^2 + 1))/(x + 1)^(2/3)#?
1 Answer
Aug 23, 2017
Explanation:
We let
#y = (xsqrt(x^2 + 1))/(x + 1)^(2/3)#
Now using logarithmic differentiation, we have:
#lny = ln((xsqrt(x^2 + 1))/(x + 1)^(2/3))#
We can now use laws of logarithms to simplify.
#lny = ln(x) + ln(x^2 + 1)^(1/2) - ln(x + 1)^(2/3)#
#lny = ln(x) + 1/2ln(x^2 + 1) - 2/3ln(x + 1)#
Now differentiate term by term .
#1/y(dy/dx) = 1/x + (2x)/(2(x^2 + 1)) - 2/(3(x+ 1))#
#dy/dx = y(1/x + x/(x^2 + 1) - 2/(3(x + 1)))#
#dy/dx = (xsqrt(x^2 + 1))/(x + 1)^(2/3)(1/x + x/(x^2 + 1) - 2/(3(x + 1)))#
Hopefully this helps!