Question

How do you implicitly differentiate \[ - 1 = - {y^2}x - 2xy - xy{e^x}\] ?

Answer 1

Hint: The above equation is an implicit equation. We can implicitly differentiate this equation by differentiating the equation with respect to \[x\] while keeping the other variable, \[y\] , as a function of \[x\] . In this method, we will follow the chain rule for differentiating the function \[y\] as we have assumed that \[y\] is a function of \[x\] .

Formula used:
For differentiating the above implicit equation, we follow the chain rule.
\[f\] and \[g\] are both differentiable functions and \[F\left( x \right)\] is the composite function which is defined by \[F\left( x \right) = f\left( {g\left( x \right)} \right)\] . \[F\] is differentiable and \[F'\] is the product \[F'\left( x \right) = f'\left( {g\left( x \right)} \right)g'\left( x \right)\] .
In the above formula, \[f'\left( {g\left( x \right)} \right)\] is the outer differentiate function and \[g'\left( x \right)\] is the inner differentiate function.

Complete step by step solution:
The equation given to us in the above question is \[ - 1 = - {y^2}x - 2xy - xy{e^x}\]
Now we will integrate both the sides of the equation with respect to \[x\] . We get,
\[0 = \left[ { - {y^2}\dfrac{{d\left( x \right)}}{{dx}} - x\dfrac{{\left( {d\left( {{y^2}} \right)} \right)}}{{dx}}} \right] - \left[ {2x\dfrac{{d\left( y \right)}}{{dx}} + 2y\dfrac{{d\left( x \right)}}{{dx}}} \right] - \left[ {xy\dfrac{{d\left( {{e^x}} \right)}}{{dx}} + x{e^x}\dfrac{{d\left( y \right)}}{{dx}} + {e^x}y\dfrac{{d\left( x \right)}}{{dx}}} \right]\]
\[ \Rightarrow 0 = \left[ { - {y^2}\left( 1 \right) - x\left( {2y} \right)\dfrac{{dy}}{{dx}}} \right] - \left[ {2x\dfrac{{dy}}{{dx}}\left( 1 \right) + 2y} \right] - \left[ {xy{e^x} + x{e^x}\dfrac{{dy}}{{dx}}\left( 1 \right) + y{e^x}\left( 1 \right)} \right]\]
On further evaluating this equation, we get,
\[0 = - {y^2} - 2xy\dfrac{{dy}}{{dx}} - 2x\dfrac{{dy}}{{dx}} - 2y - xy{e^x} - x{e^x}\dfrac{{dy}}{{dx}} - y{e^x}\]
Now we will collect all the terms containing \[\dfrac{{dy}}{{dx}}\] on one side of the equation.
\[ - 2xy\dfrac{{dy}}{{dx}} - 2x\dfrac{{dy}}{{dx}} - x{e^x}\dfrac{{dy}}{{dx}} = {y^2} + 2y + xy{e^x} + y{e^x}\]
\[
   \Rightarrow \dfrac{{dy}}{{dx}}\left[ { - 2xy - 2x - x{e^x}} \right] = {y^2} + 2y + xy{e^x} + y{e^x} \\
   \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{{y^2} + 2y + xy{e^x} + y{e^x}}}{{ - 2xy - 2x - x{e^x}}} \\
 \]
Hence after implicit differentiation, the final answer for the above equation is \[\dfrac{{dy}}{{dx}} = \dfrac{{{y^2} + 2y + xy{e^x} + y{e^x}}}{{ - 2xy - 2x - x{e^x}}}\] .

Note: In the above question, we have been asked to solve the equation through implicit differentiation. We have to keep this in mind that this type of differentiation can be used only when the equation consists of two independent functions and we want to derive one with respect to another. For example, in this equation we were able to derivate \[y\] with respect to \[x\] . Chain rule will be used to derive the function \[y\].