Question

method of differentiation question........

Answer 1

find \[dy /dx\]
if
\[2^{x} + 2^{y} = 2^{x+y}\]

Answer 2

@zepdrix

Answer 3

HI!!

Answer 4

i guess you are supposed to use implicit diff right?

Answer 5

on the left you would have \[2^x\ln(2)+2^y\ln(2)y'\]

Answer 6

on the right \[2^{x+y}(1+y')\ln(2)\]

Answer 7

set them equal, solve for \(y'\)

Answer 8

fyi:
by definition
\[ e^{\ln2} = 2 \]
use that to write your problem as
\[ \left( e^{\ln2} \right)^x + \left( e^{\ln2} \right)^y = \left( e^{\ln2} \right)^{x+y} \]
which is also
\[ e^{x \ln2}+ e^{y\ln2}= e^{x\ln2+y\ln2} = e^{x\ln2} e^{y\ln2}\]

Answer 9

of course ln 2 is just a number i.e. a constant, and
\[ \frac{d}{dx }e^{a x} = e^{a x} \frac{d}{dx }(a x) = e^{a x} a\frac{dx}{dx }\\= a e^{a x} \]