Differentiate y=[1-(1/x)]/(x-1)

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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dy/dx = d/dx {[1-(1/x)] / (x-1)}

d/dx {[1-(1/x)] / (x-1)}= [(x^2-x)*d/dx(x-1) - (x-1)*d/dx(x^2-x)]/x^2*(x-1)^2

d/dx {[1-(1/x)] / (x-1)}= [x^2 - x - (x-1)(2x-1)]/x^2*(x-1)^2

We'll remove the brackets:

d/dx {[1-(1/x)] / (x-1)} = (x^2 - x - 2x^2 + x + 2x - 1)/x^2*(x-1)^2

We'll eliminate like terms:

d/dx {[1-(1/x)] / (x-1)} = -(x^2 - 2x + 1)/x^2*(x-1)^2

dy/dx = -(x-1)^2/x^2*(x-1)^2

dy/dx = - 1/x^2

william1941's profile pic

william1941 | College Teacher | (Level 3) Valedictorian

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We have to differentiate y = [1 - (1/x)] / (x-1)

Now we can start with rewriting the expression for y = [1 - (1/x)] / (x-1) so that it is easier to differentiate

Start with writing 1-(1/x) as (x-1)/x

=>  y = [(x-1) /x] / (x-1)

Now divide by numerator and denominator by (x-1)

=> y = 1/ x

Now the derivative of 1/x = x^-1 is -1*[x^ (-1-1)] = -1*x^-2 = -1/ x^2

So as y= [1 - (1/x)] / (x-1)

=> y' = -1/ x^2

The required derivative is -1/ x^2

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