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What is the derivative of # f(x) = (x^3-3)/x#?

Answer 1

Christopher Agresta

The derivative of #f(x)# is #(2x^3+3)/(x^2)#.

Using the quotient rule of derivatives:

#d/dx((f(x))/(g(x)))=(d/dx(f(x))*g(x)-f(x)*d/dx(g(x)))/((g(x))^2)#

Here's our expression:

#color(white)=d/dx((x^3-3)/x)#

#=(d/dx(x^3-3)*x-(x^3-3)*d/dx(x))/(x^2)#

#=((3x^2-0)*x-(x^3-3)*1)/(x^2)#

#=(3x^3-x^3+3)/(x^2)#

#=(2x^3+3)/(x^2)#

This is the derivative. Hope this helped!

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Answer 2

Ezra Adams

To find the derivative of ( f(x) = \frac{x^3 - 3}{x} ), use the quotient rule. The derivative is given by:

[ f'(x) = \frac{(x \cdot (3x^2) - (x^3 - 3) \cdot 1)}{x^2} ]

Simplify this expression to obtain the derivative:

[ f'(x) = \frac{3x^3 - x^3 + 3}{x^2} ] [ f'(x) = \frac{2x^3 + 3}{x^2} ]

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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.