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What is the implicit derivative of #10=xy-y^2+x^2 #?

Answer 1

Isaiah Allen

#(dy)/(dx)=(y+2x)/(x-2y)#

#10=xy-y^2+x^2#

#0=xtimes(dy)/(dx)+y-2ytimes(dy)/(dx)+2x#

#0=(x-2y)times(dy)/(dx)+y+2x#

#(dy)/(dx)=(y+2x)/(x-2y)#

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Answer 2

Isaiah Allen

To find the implicit derivative of (10 = xy - y^2 + x^2), differentiate both sides of the equation with respect to (x), treating (y) as an implicit function of (x).

[ \frac{d}{dx}(10) = \frac{d}{dx}(xy) - \frac{d}{dx}(y^2) + \frac{d}{dx}(x^2) ]

[ 0 = y + x\frac{dy}{dx} - 2y\frac{dy}{dx} + 2x ]

[ 0 = (x - 2y)\frac{dy}{dx} + y + 2x ]

[ \frac{dy}{dx} = \frac{-y - 2x}{x - 2y} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.