Can someone explain this diffrentation question?

a) y^3 + 2sqrt(xy) = 3xy^2+ y

look at each term separately ... then add them

y^3 ..... (3y^2)(dy/dx)

2 (x^(1/2))(y^(1/2)) .... use the product rule ...

..... 2(x^(1/2))(1/2)(y^(-1/2)(dy/dx)) + 2((1/2)x^(-1/2))(y^(1/2))

3xy^2 ..... 3x(2y (dy/dx)) + 3y^2

y ..... dy/dx

(3y^2)(dy/dx) + 2(x^(1/2))(1/2)(y^(-1/2)(dy/dx)) + 2((1/2)x^(-1/2))(y^(1/2)) = 3x(2y (dy/dx)) + 3y^2 + dy/dx

simplify algebraically then move all terms with dy/dy to the left

then factor dy/dx as GCF ... then divide the other term to get the form dy/dx = aaaa/bbbb

To much algebra to do on the computer screen ... I'd go batty trying to keep it all straight.

b) use a differential to approximate 3^√27.01+(1/3^√27.01)

I assume you mean cube root (27.01) + 1 / cube root (27.01)

make an equation y = cube root x = x^(1/3)

dy = (1/3) x^(-2/3) dx

x = 27 ... so y = 3 and dx = 0.01

dy = (1/3) (27)^(-2/3) (0.01) = (1/2700)

but for x = 27.01 ... y = 3 + dy = 3 + 1/2700 = 8101/2700

Back to your problem ...

... cube root (27.01) + 1 / cube root (27.01)

= 8101/2700 + 1 /(8101 / 2700)

= 8101/2700 + 2700/8101

= approx... 3.333662557