Use logaritmic differentionation to find.....

....the derivative of y in respect to x:

y= (x√(x^2+6)/(x+7)^(2/3)

oh boy. Lots of typing.

by taking the natural log of both sides, you can really slice down the problem. That's the idea of logarithmic differentiation.

ln(y) = ln(x) + (1/2)ln(x^2 + 6) - (2/3)ln(x+7)

Differentiating the left will give you y'/y on the left, then you'll need to multiply both sides by y (the original giant fraction thing) to find y'. If you don't understand how taking the natural log decomposed the fraction into smaller parts, review laws of natural logarithms.

Hope that helped.

First begin by taking the natural logarithm of both sides:

ln(y) = ln(x) + 1/2*ln(x^2 + 6) - 2/3*ln(x + 7)

Now differentiate both sides:

y'/y = 1/x + x/(x^2 + 6) - 2/3*1/(x + 7)

solve for y'.