Question

If $2^x+2^y=2^{x+y}$, then the value of $\frac{dy}{dx}$ at $(1,1)$ is equal to

• A. $-2$
• B. $-1$
• C. $0$
• D. $1$
• E. $2$

Answer 1

The correct option is B $-1$

We have $2^x+2^y=2^{x+y}$

On differentiating w.r.t $x$

$2^x\log 2+2^y\log 2\frac{dy}{dx}=2^{x+y}\log 2(1+\frac{dy}{dx})$

$\Rightarrow 2^x+2^y\frac{dy}{dx}=2^{x+y}(1+\frac{dy}{dx})$

$\Rightarrow \frac{dy}{dx}=\frac{2^x-2^{x+y}}{2^{x+y}-2^y}$

$\therefore {\left(\frac{dy}{dx}\right)}_{(1,1)}=\frac{2-4}{4-2}=1$