Differentiation Question 14
Question: If $ 2^{x}+2^{y}={2^{x+y}}, $ then the value of $ \frac{dy}{dx} $ at $ x=y=1 $ is
[Karnataka CET 2000]
Options:
A) 0
B) - 1
C) 1
D) 2
Show Answer
Answer:
Correct Answer: B
Solution:
$ 2^{x}+2^{y}={2^{x+y}} $ ; Differentiating w.r.t. x, we get $ 2^{x}(\log 2)+2^{y}(\log 2)\frac{dy}{dx} $ = $ {2^{(x+y)}}.(\log 2)( 1+\frac{dy}{dx} ) $
Therefore $ 2^{x}+2^{y}\frac{dy}{dx}={2^{x+y}}+{2^{x+y}}( \frac{dy}{dx} ) $
Therefore $ \frac{dy}{dx}(2^{y}-{2^{x+y}})={2^{x+y}}-2^{x} $
Therefore $ \frac{dy}{dx}=\frac{{2^{x+y}}-2^{x}}{2^{y}-{2^{x+y}}} $ .
$ \therefore {{( \frac{dy}{dx} )} _{x=y=1}}=\frac{2^{2}-2}{2-2^{2}}=\frac{2}{-2}=-1. $
BETA
