How do you find the derivative of #h(x) = (x^(1/3)) / (x^2+1)#?

Answer 1

#1/(3x^(2/3)(x^2 +1)) - (2x^(4/3))/ ((x^2 +1)^2#

With #h(x) = (x^(1/3))/(x^2+1)# you use the quotient rule, which states:
#(f/g)'= (g f' - f g')/ g^2 #
Derivative of #f(x)=x^(1/3)#:
#(1/3)x^(-2/3)#
#f'(x) = 1/(3x^(2/3))#
Derivative of #g(x)=(x^2+1)#:
#g'(x)=2x#

Now connect the plug:

#((x^2+1)*1/(3x^(2/3)) - (x^(1/3)*2x))/(x^2+1)^2#
#((x^2+1)/(3x^(2/3)) - 2x^(4/3))/ ((x^2 +1)^2#
#((x^2+1)/(3x^(2/3)))/(x^2 +1)^2 - (2x^(4/3))/ ((x^2 +1)^2#
#1/(3x^(2/3)(x^2 +1)) - (2x^(4/3))/ ((x^2 +1)^2#
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Answer 2

To find the derivative of ( h(x) = \frac{x^\frac{1}{3}}{x^2+1} ), use the quotient rule. The quotient rule states that if you have a function ( f(x) = \frac{g(x)}{h(x)} ), then the derivative is given by ( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} ). Apply this rule to ( h(x) ), where ( g(x) = x^\frac{1}{3} ) and ( h(x) = x^2 + 1 ).

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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.

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.

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.

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.

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