How do you differentiate #f(x)= e^x/(e^(3x) +2x )# using the quotient rule?
To differentiate a quotient, you must use the quotient rule if the function cannot be simplified.
To differentiate a quotient, use the following:
To remember this you can use the "mnemonic":
"Low di hi minus hi di low all over low squared" where "di hi" or "di low" means to differentiate the top or the bottom respectively.
Using this strategy we get:
And that's it! Hopefully this was clear and concise! Should you have any questions, please feel free to ask! :)
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To differentiate ( f(x) = \frac{e^x}{e^{3x} + 2x} ) using the quotient rule, follow these steps:

Identify the numerator and denominator functions: ( u(x) = e^x ) ( v(x) = e^{3x} + 2x )

Compute the derivatives of ( u(x) ) and ( v(x) ): ( u'(x) = e^x ) ( v'(x) = e^{3x} + 2 )

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

Substitute the derivatives and functions into the formula: ( f'(x) = \frac{(e^x)(e^{3x} + 2x)  (e^{3x} + 2)(e^x)}{[e^{3x} + 2x]^2} )

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

Combine like terms in the numerator: ( f'(x) = \frac{2e^xe^{3x} + 2xe^x  2e^x}{[e^{3x} + 2x]^2} )

Factor out common terms: ( f'(x) = \frac{2e^x(e^{3x} + x  1)}{[e^{3x} + 2x]^2} )
This is the derivative of the given function ( f(x) ) using the quotient rule.
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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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