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How do you find #dy/dx# by implicit differentiation given #xy^3=y+x#?

Answer 1

Evelyn Agard

#dy/dx=(1-y^3)/(3xy^2-1)#

differentiate each term on both sides #color(blue)"implicitly with respect to x"#

#"Note differentiate " xy^3" using "color(blue)"product rule"#

#(x.3y^2.dy/dx+y^3 .1)=dy/dx+1#

#rArr3xy^2dy/dx-dy/dx=1-y^3#

#rArrdy/dx(3xy^2-1)=1-y^3#

#rArrdy/dx=(1-y^3)/(3xy^2-1)#

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Answer 2

Paisley Johnson

To find dy/dx by implicit differentiation for the equation xy^3 = y + x, differentiate both sides of the equation with respect to x. Then, solve for dy/dx.

Differentiating both sides with respect to x gives:

(1)(y^3) + (x)(3y^2(dy/dx)) = (dy/dx) + 1

Simplify and solve for dy/dx.

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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.