how do i differentiate the following?

Question

3(squarerootsign) x +2 / (squarerootsign) x

x is the letter x

in the question it may help to say it is the cube root of x

Answer 1

Let us first put it in a better form
3((x+2)/x)^(1/2) or 3((1+2/x )^(1/2)

Then differentiate the function times the derivative of the argument and we have
3((1/2)(((x+2)/x)^(3/2)(-2/(x^2))

Answer 2

You differentiate each term before and after the '+' seperately. The first will be 3 * x(raised to the power 1/2), since square root and raised to the power half are the same thing. The differential of that will follow the formula d/dx(x(raised to the power n))= n * x(raised to the power n-1). Same goes for the second part of the equation where the power will be (-1/2). Since 2 and 3 are constants they can be taken out of the differential.
For example the differential of 3 * x ( raised to the power 5/2)
= 3 * d/dx(x ^(5/2))= 3*(5/2)*x((5/2)-1)
= 15/2 * x^(3/2)

Answer 3

The derivative of a sum is the sum of the derivatives.

Derivative of 3sqrt(x): 3/(2sqrt(x))
Derivative of 2/sqrt(x): well, you write 2/sqrt(x)=2*x^(-1/2). Now differentiate: you get -x(-3/2)=-1/(x*sqrt(x))

So the answer is: 3/2sqrt(x) - 1/(x*sqrt(x))

Answer 4

EDIT: I misread your problem. I wasn't clear just exactly what was under the first radical.

Answer 5

3/(2x^1/2)-1/(x^3/2)

Answer 6

You don't...