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Calculus/Implicit Differentiation

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Expert: - 4/2/2009


Question
If x + y = xy, then dy/dx is

a. 1/(x-1)
b. (y-1)/(x-1)
c. (1-y)/(x-1)
d. x + y - 1
e. (2-xy)/y

Answer
To take the derivative of x + y = xy, note that y is a function of x.

As such, the derivative of x is 1.

The derivative of y is y'.

The value of xy is differentiatied by the product rule.
That is, xy' + yx', where x' is 1.  So the value is xy' + y.

The total equation si 1 + dy/dx = x(dy/dx) + y.

Subtract the y from both sides of the equation and also subtract dy/dx from both sides.  This gives

1 - y = x(dy/dx) - dy/dx.

Factor out the dy/dx, giving

1 - y = (dy/dx)(x-1).

Divide both sides by x-1, and then you get the answer.

It is (1-y)/(x-1) = dy/dx.

Look at the choices.  It's there.

Calculus

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