Question

How do you differentiate \[x + y = xy\]?

Answer 1

Hint: Derivative are defined as the varying rate of a function with respect to an independent variable. To differentiate the right hand side of the equation we use the product rule. That is if we have \[y = uv\] then its differentiation with respect to ‘x’ is \[\dfrac{{dy}}{{dx}} = u \times \dfrac{{dv}}{{dx}} + v \times \dfrac{{du}}{{dx}}\]. We solve this using implicit differentiation. We know that the differentiation of ‘x’ with respect to ‘x’ is 1.

Complete step-by-step solution:
Given, \[x + y = xy\].
Now differentiate implicitly with respect to ‘x’.
\[\dfrac{d}{{dx}}\left( {x + y} \right) = \dfrac{d}{{dx}}\left( {xy} \right)\]
Applying the product rule \[\dfrac{{dy}}{{dx}} = u \times \dfrac{{dv}}{{dx}} + v \times \dfrac{{du}}{{dx}}\] in the right hand side of the differential equation, Where \[u = x\] and \[v = y\], then
\[\dfrac{d}{{dx}}(x) + \dfrac{d}{{dx}}(y) = x \times \dfrac{d}{{dx}}(y) + y\dfrac{d}{{dx}}(x)\]
\[1 + \dfrac{{dy}}{{dx}} = x.\dfrac{{dy}}{{dx}} + y\]
Grouping \[\dfrac{{dy}}{{dx}}\] on one side we have,
\[\dfrac{{dy}}{{dx}} - x.\dfrac{{dy}}{{dx}} = y - 1\]
Taking \[\dfrac{{dy}}{{dx}}\] as common we have,
\[\dfrac{{dy}}{{dx}}\left( {1 - x} \right) = y - 1\]
Divide the whole differential equation by \[(x - 1)\], we have,
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{y - 1}}{{1 - x}}\].
Thus the differentiation of \[x + y = xy\] is \[\dfrac{{y - 1}}{{1 - x}}\].

Note: We know the differentiation of \[{x^n}\] with respect to ‘x’ is \[\dfrac{{d({x^n})}}{{dx}} = n.{x^{n - 1}}\]. We know the differentiation of \[{y^n}\] with respect to ‘x’ is \[\dfrac{{d({y^n})}}{{dx}} = n.{y^{n - 1}}\dfrac{{dy}}{{dx}}\].

We also have different rules in the differentiation. Those are
i) Linear combination rule: The linearity law is very important to emphasize its nature with alternate notation. Symbolically it is specified as \[h'(x) = af'(x) + bg'(x)\]

ii) Quotient rule: The derivative of one function divided by other is found by quotient rule such as\[{\left[ {\dfrac{{f(x)}}{{g(x)}}} \right]’} = \dfrac{{g(x)f’(x) - f(x)g’(x)}}{{{{\left[ {g(x)} \right]}^2}}}\].

iii) Product rule: When a derivative of a product of two function is to be found, then we use product rule that is \[\dfrac{{dy}}{{dx}} = u \times \dfrac{{dv}}{{dx}} + v \times \dfrac{{du}}{{dx}}\].

iv) Chain rule: To find the derivative of composition function or function of a function, we use chain rule. That is \[fog'({x_0}) = [(f'og)({x_0})]g'({x_0})\]. We use these rules depending on the given problem.