How do you find the differential #dy# of the function #y=sqrtx+1/sqrtx#?

Answer 1

Note that #1/sqrtx = x^(-1/2)#, and #sqrt x = x^(1/2)#, then use the power rule to obtain #dy/(dx)=1/2x^(-1/2)-1/2x^(z3/2)#. Then multiply both sides by dx to get #dy=(1/2x^(-1/2)-1/2x^(z3/2))dx#.

It will be presumed here that you meant exactly what you said, and want the differential #dy# rather than the derivative #dy/(dx)#. If you instead want the derivative, simply ignore the last step.
Recall the power rule states that for any term #ax^b#, with a and b as constants, the derivative is #dy/(dx) = (b)ax^(b-1)#. Recall further that #sqrtx = x^(1/2)# and that #1/sqrtx = 1/x^(1/2) = x^(-1/2)# (because any term #p/x^q = px^(-q)# by the properties of exponents). Thus, if given the function above, we can find its derivative (and differential) as follows:
#y =sqrtx + 1/sqrtx= x^(1/2)+x^(-1/2)#.
#-> dy/(dx) = d/(dx)x^(1/2)+ d/(dx)x^(-1/2) = 1/2x^(-1/2) - 1/2x^(-3/2)#
#dy/(dx)= 1/2x^(-1/2) - 1/2x^(-3/2)#
This, we now have the derivative. If we want the differential #dy#...
#dy/(dx)= 1/2x^(-1/2) - 1/2x^(-3/2) -> dy = ( 1/2x^(-1/2) - 1/2x^(-3/2))dx#
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Answer 2

To find the differential ( dy ) of the function ( y = \sqrt{x} + \frac{1}{\sqrt{x}} ), differentiate the function with respect to ( x ) and then multiply by ( dx ).

[ y = \sqrt{x} + \frac{1}{\sqrt{x}} ]

[ \frac{dy}{dx} = \frac{d}{dx}(\sqrt{x}) + \frac{d}{dx}\left(\frac{1}{\sqrt{x}}\right) ]

Using the power rule and the chain rule, differentiate each term:

[ \frac{dy}{dx} = \frac{1}{2\sqrt{x}} - \frac{1}{2x\sqrt{x}} ]

[ dy = \left(\frac{1}{2\sqrt{x}} - \frac{1}{2x\sqrt{x}}\right) dx ]

This is the differential of the function ( y = \sqrt{x} + \frac{1}{\sqrt{x}} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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