How do you differentiate #f(x) =x/(e^(3x)+x^3)# using the quotient rule?
Adding each of these findings to Equation (1):
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To differentiate ( f(x) = \frac{x}{e^{3x} + x^3} ) using the quotient rule, follow these steps:

Identify the numerator and denominator functions. ( f(x) = u(x)/v(x) ), where ( u(x) = x ) and ( v(x) = e^{3x} + x^3 ).

Apply the quotient rule formula: ( f'(x) = \frac{u'(x)v(x)  u(x)v'(x)}{(v(x))^2} ).

Find the derivatives of the numerator and denominator functions: ( u'(x) = 1 ) and ( v'(x) = e^{3x} + 3x^2 ).

Substitute the derivatives and the original functions into the quotient rule formula: ( f'(x) = \frac{(1)(e^{3x} + x^3)  (x)(e^{3x} + 3x^2)}{(e^{3x} + x^3)^2} ).

Simplify the expression: ( f'(x) = \frac{e^{3x} + x^3 + xe^{3x}  3x^3}{(e^{3x} + x^3)^2} ).

Further simplify if necessary.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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