# What is the implicit derivative of #1= xy-e^(xy) #?

first term, using product rule:

second term using chain rule:

combining terms

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To find the implicit derivative of ( 1 = xy - e^{xy} ), we differentiate both sides of the equation with respect to ( x ) using implicit differentiation.

( \frac{d}{dx}(1) = \frac{d}{dx}(xy - e^{xy}) )

( 0 = y + x\frac{dy}{dx} - e^{xy}(y + x\frac{dy}{dx}) )

( 0 = y + x\frac{dy}{dx} - ye^{xy} - x(e^{xy})\frac{dy}{dx} )

( 0 = y + x\frac{dy}{dx} - ye^{xy} - xe^{xy}\frac{dy}{dx} )

Now, we solve for ( \frac{dy}{dx} ):

( x\frac{dy}{dx} - xe^{xy}\frac{dy}{dx} = ye^{xy} - y )

( \frac{dy}{dx}(x - xe^{xy}) = y(e^{xy} - 1) )

( \frac{dy}{dx} = \frac{y(e^{xy} - 1)}{x - xe^{xy}} )

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