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Question: What is the implicit derivative of \(1={{e}^{xy}}\)?...

What is the implicit derivative of 1=exy1={{e}^{xy}}?

Explanation

Solution

To solve the above question we should know about the implicit differentiation. Actually implicit differentiation is the procedure of differentiating an implicit equation with respect to the desired variable while treating the other variables as unspecified functions of x. To differentiate an implicit function, any of the following methods is followed:
1 In the first method the implicit equation is solved for y and it is expressed explicitly in terms of x and differentiation of y is carried. This method is found useful only when y is easily expressible in terms of x.
2 In second, Y is thought of as a function of x, and both members of the implicit equation are differentiating with respect to x. The resulting equation is solved to find the value of dydx\dfrac{dy}{dx} .

Complete step by step solution:
The given implicit equation is:
⇒1=exy\Rightarrow 1={{e}^{xy}}
To find the derivative of the given equation, we will use the second method to find the value of dydx\dfrac{dy}{dx}.
Now to differentiate the given equation we have to use chain rule. The formula to find derivative of any function by chain rule is:
⇒d(u.v)dx=u ˋv+v ˋu\Rightarrow \dfrac{d\left( u.v \right)}{dx}={{u}^{\grave{\ }}}v+{{v}^{\grave{\ }}}u
Now the given equation is 1=exy1={{e}^{xy}}, the left side of the equation is constant 1 so its derivative with respect to x is zero, the by applying chain rule we get
⇒0=exy[y+xdydx] ⇒0=yexy+xexydydx \begin{aligned} & \Rightarrow 0={{e}^{xy}}\left[ y+x\dfrac{dy}{dx} \right] \\\ & \Rightarrow 0=y{{e}^{xy}}+x{{e}^{xy}}\dfrac{dy}{dx} \\\ \end{aligned}
Now we have to find dydx\dfrac{dy}{dx}so we will separate dydx\dfrac{dy}{dx}from the above expression, we get
⇒−yexy=dydxxexy ⇒dydx=−yexyxexy ⇒dydx=−yx \begin{aligned} & \Rightarrow -y{{e}^{xy}}=\dfrac{dy}{dx}x{{e}^{xy}} \\\ & \Rightarrow \dfrac{dy}{dx}=\dfrac{-y{{e}^{xy}}}{x{{e}^{xy}}} \\\ & \Rightarrow \dfrac{dy}{dx}=\dfrac{-y}{x} \\\ \end{aligned}

Hence we the implicit derivative of the given equation 1=exy1={{e}^{xy}}is dydx=−yx\dfrac{dy}{dx}=\dfrac{-y}{x}.

Note: To solve these types of question first examine the question carefully and then apply the first and second methods which we discussed above. To solve the implicit equation we have to treat one function as a constant. So in the process of finding the derivative of an implicit function we have to remember which function is constant.