y = x*(x-2)^-1/2. Prove dy\dx = (x-4)/2*(x-2)^3/2

Firstly, when approaching a differentiation question you need to work out what method you need to use to solve it. As you can see there are two terms multiplied by one another (the 'x' term and the '(x-2)-1/2' term), therefore the product rule must be used. 

Making u = x and v = (x-2)-1/2

du/dx = 1 dv/dx = -1/2*(x-2)-3/2

Substituing these things into the Product Rule equation we get: 

dy/dx = -x/2*(x-2)-3/2 + (x-2)-1/2

Now we need to focus on manipulating this equation to match the one given in the question. To start with we will take out a factor of (x-2)-3/2 giving: 

dy/dx = (x-2)-3/2*(-x/2 + x-2) 

Simplyfying : 

dy/dx = (x-2)-3/2* (x/2 -2)

Multiplying by 2 : 

dy/dx = (x-4)/2*(x-2)3/2

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