Question

Find the derivative of the following \[x{{e}^{y}}+1=xy\]

Answer 1

Hint: To solve the above problem we have to know the basic derivatives of functions. To solve further we have to use the uv rule. If u and v are two functions of x, then the derivative of the product uv is given by \[\dfrac{d\left( uv \right)}{dx}=u\dfrac{dv}{dx}+v\dfrac{du}{dx}\].

Complete step-by-step answer:
Given \[x{{e}^{y}}+1=xy\]
Differentiating both sides of the equation we get,
\[\dfrac{d}{dx}\left( x{{e}^{y}}+1 \right)=\dfrac{d}{dx}\left( xy \right)\]
Now taking the L.H.S term,
\[\dfrac{d}{dx}\left( x{{e}^{y}}+1 \right)\]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a)
\[\dfrac{d}{dx}\left( x \right)=1\] . . . . . . . . . . . . . . . . . . . . . . . . . (1)
\[\dfrac{d}{dx}\left( {{e}^{x}} \right)={{e}^{x}}\] . . . . . . . . . . . . . . . . . . . . . . . . (2)
By differentiating the above term (a) we get
\[={{e}^{y}}+x{{e}^{y}}\dfrac{d}{dx}\left[ y \right]\]
Now taking R.H.S term,
\[\dfrac{d}{dx}\left( xy \right)\]
We have seen hints how the uv rule works.
\[\dfrac{d\left( uv \right)}{dx}=u\dfrac{dv}{dx}+v\dfrac{du}{dx}\]
\[=y+x\dfrac{d}{dx}\left[ y \right]\]
Reforming the equation by setting the L.H.S =R.H.S
\[{{e}^{y}}+x{{e}^{y}}\dfrac{d}{dx}\left[ y \right]\]\[=y+x\dfrac{d}{dx}\left[ y \right]\]
Further solving for \[\dfrac{dy}{dx}\]
\[x{{e}^{y}}\dfrac{d}{dx}\left[ y \right]-x\dfrac{d}{dx}\left[ y \right]=y-{{e}^{y}}\]
\[(x{{e}^{y}}-x)\dfrac{d}{dx}\left[ y \right]=y-{{e}^{y}}\]
\[\dfrac{dy}{dx}=\dfrac{y}{(x{{e}^{y}}-x)}-\dfrac{{{e}^{y}}}{(x{{e}^{y}}-x)}\]

Note: In the above problem we have used the product rule. After using this rule we have got the values for \[\dfrac{dy}{dx}\]. Further solving for \[\dfrac{dy}{dx}\]made us towards a solution. If we are doing derivative means we are finding the slope of a function. Care should be taken while doing calculations.