How do you differentiate #f(x)= 1/(e^(3x) 5x)# using the quotient rule?
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To differentiate ( f(x) = \frac{1}{e^{3x}  5x} ) using the quotient rule, let ( u(x) = 1 ) and ( v(x) = e^{3x}  5x ).
Then, applying the quotient rule ( \left( \frac{u}{v} \right)' = \frac{u'v  uv'}{v^2} ), where ( u' ) and ( v' ) denote the derivatives of ( u ) and ( v ) respectively:
[ u'(x) = 0 ] [ v'(x) = 3e^{3x}  5 ]
[ f'(x) = \frac{0 \cdot (e^{3x}  5x)  1 \cdot (3e^{3x}  5)}{(e^{3x}  5x)^2} ] [ f'(x) = \frac{3e^{3x} + 5}{(e^{3x}  5x)^2} ]
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To differentiate ( f(x) = \frac{1}{{e^{3x}  5x}} ) using the quotient rule:

Identify the numerator and denominator of the function ( f(x) ):
 Numerator: 1
 Denominator: ( e^{3x}  5x )

Apply the quotient rule, which states that for a function ( \frac{u(x)}{v(x)} ), the derivative is given by: [ \frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x)  u(x)v'(x)}{(v(x))^2} ]

Differentiate the numerator and denominator separately:
 ( u(x) = 1 ), so ( u'(x) = 0 )
 ( v(x) = e^{3x}  5x ), so ( v'(x) = 3e^{3x}  5 )

Apply the quotient rule formula to find ( \frac{d}{dx} \left( \frac{1}{{e^{3x}  5x}} \right) ): [ \frac{d}{dx} \left( \frac{1}{{e^{3x}  5x}} \right) = \frac{0 \cdot (e^{3x}  5x)  1 \cdot (3e^{3x}  5)}{(e^{3x}  5x)^2} ]

Simplify the expression: [ \frac{3e^{3x} + 5}{{(e^{3x}  5x)^2}} ]
Therefore, the derivative of ( f(x) = \frac{1}{{e^{3x}  5x}} ) using the quotient rule is ( \frac{3e^{3x} + 5}{{(e^{3x}  5x)^2}} ).
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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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