Differentiate with respect to x y = (5 - 1/x)/(x - 1)

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neela | High School Teacher | (Level 3) Valedictorian

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To differentiate with respect to x:

y = (5-1/x)/(x-1).

y = 5/(x-1) - 1/[x(x-1)]

 y = 5/(x-1) - (1/(x-1) + 1/x)

y = 5/(x-1) - 1/(x-1) +1/x

 y = 4/(x-1) +1/x

Now we differentiate both sides:

dy/dx = d/dx(4/(x-1)) + d/dx(1/x).

dy/dx = 4 d/dx (x-1)^(-1) + d/dx (x^(-1).

 dy/dx = 4(-1)(x-1)^(-1-1)  + (-1) x^(-1-1) , as d/dx(x^n) = n x^(n-1).

dy/dx = -4(x-1)^-2 - x^-2

dy/dx = -{4/(x-1)^2 +1/x^2}

 

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll differentiate with respect to x:

dy/dx = d/dx {[5-(1/x)] / (x-1)}

dy/dx = d/dx [5/(x-1)] - d/dx [1/x(x-1)]

d/dx [5/(x-1)] = [(x-1)*d/dx(5) - 5*d/dx(x-1)]/(x-1)^2

d/dx [5/(x-1)] = [0*(x-1) - 5*1]/(x-1)^2

d/dx [5/(x-1)] = - 5/(x-1)^2 (1)

 

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

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

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

dy/dx = (1) - (2)

dy/dx = - 5/(x-1)^2 + (2x-1)/x^2*(x-1)^2

dy/dx = (2x - 1 - 5x^2)/x^2*(x-1)^2

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