find derivative of (8sqrt(x)+(9/2root3(x)))^2
You have not supplied sufficient parentheses to render the expression unambiguous, that probably explains why you get different results from different sources.
Is it
(9/2)root3(x)
or
9/(2root3(x)) ?
its like 9/2(x)^1/3
It is still not clear to me if (x)^1/3 is with the numerator or the denominator.
I assume you are transcribing from a type-set fraction where the paretheses around the denominator are understood. When transcribing to a single line (as in this case), you will need to insert explicitely the parentheses around ALL denominators and numerators to avoid ambiguity.
its in denominator
So we're looking at find the derivative of the following expression:
y=(8*sqrt(x)+((9/2)/(x)^(1/3)))^2
Using the chain rule, we get
dy/dx
= 2(8*sqrt(x)+((9/2)/(x)^(1/3))) * d(8*sqrt(x)+((9/2)/(x)^(1/3)))/dx
By writing
=(8*sqrt(x)+((9/2)/(x)^(1/3)))^2
=8x^(1/2) + (9/2)x^(-1/3)
We can differentiate u using the power rule:
du/dx = 8(1/2)x^(-1/2) + (9/2)(-1/3)x^(-4/3)
So dy/dx
=2(8*sqrt(x)+((9/2)/(x)^(1/3)))*(8(1/2)x^(-1/2) + (9/2)(-1/3)x^(-4/3) )
=64+12x^(5/6)-(27/2)x^(-5/3)
after simplification.
Check my work.