Given y=rootx + 4/rootx = 4, find the value of dy/dx when x=8, writing your answer in the form aroot2, where a is a rational number.

First, we will write out the equation, writing the root x values in a way that allows us to differentiate them. Root x can be written as x^1/2 and 4/rootx can be written as 4x^-1/2. Now we can differentiate. Remember: to differentiate we times the x constant by the value of the power to which x is raised, and then subtract one from the power. In this case x^1/2 will differentiate to 1/2(x)^-1/2 and 4/root x will differentiate to -2x^-3/2, and we can ignore the 4 since it is not attached to an x constant. Therefore dy/dx = 1/2(x)^-1/2 - 2x^-3/2, or we can simplify it to dy/dx= 1/2rootx - 2/(rootx)^3.
Now we can substitute in our value for x ,which is 8. So when x = 8, dy/dx = 1/2root8 - 2/(root8)^3. Remembering our laws of surds, root8^3 is 8root8. Furthermore, we can write root8 as 2root2. So we now have 1/4root2 - 2/16root2. Now, in order to simplify the answer, we must subtract these fractions from one another, and to do so we must have a common denominator. If we multiply the whole of the first fraction by 4 we will have a common denominator of 16root2 and can complete the sum, leaving us with 2/16root2. However, the question asks us to leave our answer in the form aroot2, so we can't leave the root 2 on the bottom of the fraction. If we multiply the whole equation by root 2 we will get 2root2/32, which can be simplified to 1/16(root2) or root2/16. And there is our answer.