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How do you differentiate #x^4(x+y)=y^2(3x-y)#?

Answer 1

First, use the distributive property to write #x^5+x^4y=3xy^2-y^3#.

Next, make the assumption that #y# is a function of #x# and differentiate both sides with respect to #x#, using the Product Rule and Chain Rule as necessary:

#5x^4+4x^3y+x^4 dy/dx=3y^2+6x y dy/dx-3y^2 dy/dx#

Finally, solve for #dy/dx#:

#dy/dx(x^4-6xy+3y^2)=-5x^4-4x^3y+3y^2#

#dy/dx=\frac{-5x^4-4x^3y+3y^2}{x^4-6xy+3y^2}#

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Answer 2

Charles Arocha

To differentiate the equation (x^4(x+y)=y^2(3x-y)) with respect to (x), you would use the product rule and chain rule.

Here's the step-by-step process:

Expand both sides of the equation.
Differentiate each term with respect to (x).
Simplify the resulting expression.

The solution would involve applying the product rule and chain rule to each term, followed by simplification.

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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.