Question

If $e^x+e^y=e^{(x+y)}$, then $\frac{dy}{dx}at(2,2)$ is

Answer 1

Step $1$ : Differentiating both sides with respect to $x$

Given that $e^x+e^y=e^{(x+y)}$

$$\begin{gathered} \Rightarrow e^x+\left(e^y\right)\left(\frac{dy}{dx}\right)=e^{x+y}(1+\frac{dy}{dx}) \\ \Rightarrow e^x–e^{x+y}=\left(\frac{dy}{dx}\right)(e^{x+y}–e^y) \\ \Rightarrow \left(\frac{dy}{dx}\right)=\frac{(e^{x+y}–e^{x^{}})}{(e^y–e^{x+y})} \end{gathered}$$

Step $2$ :Finding the value of $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{}\mathit{a}\mathit{t}\mathbf{}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{}\mathbf{2}\mathbf{)}\mathbf{:}$

$$\begin{gathered} \Rightarrow \left(\frac{dy}{dx}\right)=\frac{(e^{2+2}–e^2)}{(e^2–e^{2+2})} \\ \Rightarrow =\frac{-(e^{2+2}–e^2)}{(e^{2+2}-e^2)} \\ \Rightarrow =-1 \end{gathered}$$

Hence, the value of $\frac{dy}{dx}at(2,2)$ is $\mathbf{-}\mathbf{1}$.