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How do you find #dy/dx# by implicit differentiation given #x^4+y^4=5#?

Answer 1

Isabella Ackerman

The answer is #=-x^3(5-x^4)^(-3/4)#

Let's do the differentiation

#x^4+y^4=5#

#4x^3+4y^3dy/dx=0#

#dy/dx=(-4x^3)/(4y^3)#

#=-x^3/y^3#

But, #y^4=5-x^4#

#y^3=(5-x^4)^(3/4)#

Therefore,

#dy/dx=-x^3/(5-x^4)^(3/4)=-x^3(5-x^4)^(-3/4)#

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Answer 2

Charles Anctil

To find ( \frac{dy}{dx} ) by implicit differentiation for the equation ( x^4 + y^4 = 5 ), follow these steps:

Differentiate both sides of the equation with respect to ( x ).
Treat ( y ) as a function of ( x ) and use the chain rule when differentiating terms containing ( y ).
Solve the resulting equation for ( \frac{dy}{dx} ).

Differentiating both sides with respect to ( x ) yields:

[ 4x^3 + 4y^3 \frac{dy}{dx} = 0 ]

Rearranging terms to solve for ( \frac{dy}{dx} ):

[ \frac{dy}{dx} = -\frac{x^3}{y^3} ]

This is the derivative ( \frac{dy}{dx} ) for the given implicit equation.

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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.