How do you implicitly differentiate #y= y(x-y)^2 + e^(x y) #?

Answer 1

# dy/dx=[y{e^(xy)+2(x-y)}]/{1-(x-y)(x-3y)-xe^(xy)}.#

#y=y(x-y)^2+e^(xy) rArr d/dx(y)=d/dx{y(x-y)^2+e^(xy)}.#
#:. dy/dx=d/dx{y(x-y)^2}+d/dx{e^(xy)}.....(star).#

To differentiate the R.H.S., we use the Product Rule and the

Chain Rule :

#d/dx{y(x-y)^2}=yd/dx(x-y)^2+(x-y)^2d/dx(y),#
#=y{2(x-y)}d/dx(x-y)+(x-y)^2dy/dx,#
#=2y(x-y){d/dx(x)-d/dx(y)}+(x-y)^2dy/dx,#
#=2y(x-y){1-dy/dx}+(x-y)^2dy/dx,#
#=2y(x-y)+(x-y)^2dy/dx-2y(x-y)dy/dx,#
#=2y(x-y)+(x-y){(x-y)-2y}dy/dx,#
#=2y(x-y)+(x-y)(x-3y)dy/dx.............(1).#
Similarly, #d/dx{e^(xy)}=e^(xy)d/dx{xy},#
#=e^(xy){xd/dx(y)+yd/dx(x)},#
#=e^(xy){xdy/dx+y}=xe^(xy)dy/dx+ye^(xy).............(2).#
Altogether, by #(star), (1) and (2),#
#dy/dx=2y(x-y)+(x-y)(x-3y)dy/dx+xe^(xy)dy/dx+ye^(xy)#
# :. {1-(x-y)(x-3y)-xe^(xy)}dy/dx=2y(x-y)+ye^(xy)#
#rArr dy/dx=[y{e^(xy)+2(x-y)}]/{1-(x-y)(x-3y)-xe^(xy)}.#

Enjoy Maths.!

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Answer 2

To implicitly differentiate ( y = (x - y)^2 + e^{xy} ) with respect to ( x ), you apply the chain rule and product rule where necessary.

[ \begin{align*} y &= (x - y)^2 + e^{xy} \ \frac{dy}{dx} &= \frac{d}{dx}((x - y)^2) + \frac{d}{dx}(e^{xy}) \ &= 2(x - y) \cdot \frac{d}{dx}(x - y) + e^{xy} \cdot \frac{d}{dx}(xy) \ &= 2(x - y) \cdot (1 - \frac{dy}{dx}) + e^{xy} \cdot (x \cdot \frac{dy}{dx} + y) \ &= 2(x - y) - 2(x - y)\frac{dy}{dx} + xe^{xy}\frac{dy}{dx} + e^{xy}y \end{align*} ]

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

[ \begin{align*} \frac{dy}{dx} &= \frac{2(x - y) - e^{xy}y}{2(x - y) + xe^{xy}} \ &= \frac{2x - 2y - e^{xy}y}{2x - 2y + xe^{xy}} \end{align*} ]

This is the implicit derivative of ( y ) with respect to ( x ).

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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.

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.

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.

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.

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