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How do you find the differential #dy# of the function #y=3x^(2/3)#?

Answer 1

Paisley Johnson

The differential of #y# is the derivative of the function times the differential of #x#

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

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Answer 2

Emilia Aguilar

To find the differential ( dy ) of the function ( y = 3x^{2/3} ), you can use the differentiation rules for power functions. First, differentiate the function with respect to ( x ), and then multiply by ( dx ).

The derivative of ( y ) with respect to ( x ) is found using the power rule, which states that the derivative of ( x^n ) is ( nx^{n-1} ), where ( n ) is a constant.

Applying the power rule to ( y = 3x^{2/3} ), we get:

[ \frac{dy}{dx} = \frac{d}{dx} (3x^{2/3}) = 3 \cdot \frac{2}{3} x^{(2/3) - 1} ]

[ \frac{dy}{dx} = 2x^{-1/3} ]

Multiplying both sides by ( dx ), we obtain the differential ( dy ):

[ dy = 2x^{-1/3} 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.