Applications of Derivatives Question - None

Question

If $2^x + 2^y = 2^{x+y}$, then $\frac{dy}{dx}$ is equal to

Accepted Answer

Given: $2^x + 2^y = 2^{x+y}$

Differentiating both sides with respect to $x$:

$2^x\ln 2 + 2^y\ln 2 \cdot \frac{dy}{dx} = 2^{x+y}\ln 2\left(1 + \frac{dy}{dx}\right)$

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

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

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

$\frac{dy}{dx} = \frac{2^{x+y} - 2^x}{2^y - 2^{x+y}} = \frac{1 - 2^y}{2^x - 1}$

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