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OpenStudy (purpleshadow):
method of differentiation question........
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OpenStudy (purpleshadow):
find \[dy /dx\] if \[2^{x} + 2^{y} = 2^{x+y}\]
OpenStudy (purpleshadow):
@zepdrix
OpenStudy (misty1212):
HI!!
OpenStudy (misty1212):
i guess you are supposed to use implicit diff right?
OpenStudy (misty1212):
on the left you would have \[2^x\ln(2)+2^y\ln(2)y'\]
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OpenStudy (misty1212):
on the right \[2^{x+y}(1+y')\ln(2)\]
OpenStudy (misty1212):
set them equal, solve for \(y'\)
OpenStudy (phi):
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 \[ e^{x \ln2}+ e^{y\ln2}= e^{x\ln2+y\ln2} = e^{x\ln2} e^{y\ln2}\]
OpenStudy (phi):
of course ln 2 is just a number i.e. a constant, and \[ \frac{d}{dx }e^{a x} = e^{a x} \frac{d}{dx }(a x) = e^{a x} a\frac{dx}{dx }\\= a e^{a x} \]
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