What is the derivative of (e^{3x}/(e^{3x)+2}) ?

The derivative of (e^{3x}/(e^{3x)+2}) is (6e^{3x})/((e^{3x)+2)^2}

What is the first derivative of (e^{3x}/(e^{3x)+2}) ?

The first derivative of (e^{3x}/(e^{3x)+2}) is (6e^{3x})/((e^{3x)+2)^2}

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\frac{d}{d x} (\frac{e ^{3 x}}{e ^{3 x} + 2})

\frac{6 e ^{3 x}}{( e ^{3 x} + 2 ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{e ^{3 x}}{e ^{3 x} + 2})

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( e ^{3 x} ) ( e ^{3 x} + 2 ) - \frac{d}{d x} ( e ^{3 x} + 2 ) e ^{3 x}}{( e ^{3 x} + 2 ) ^{2}}

\frac{d}{d x} (e^{3 x}) = e^{3 x} \cdot 3

\frac{d}{d x} (e^{3 x} + 2) = 3 e^{3 x}

= \frac{e ^{3 x} \cdot 3 ( e ^{3 x} + 2 ) - 3 e ^{3 x} e ^{3 x}}{( e ^{3 x} + 2 ) ^{2}}

Simplify

\frac{e ^{3 x} \cdot 3 ( e ^{3 x} + 2 ) - 3 e ^{3 x} e ^{3 x}}{( e ^{3 x} + 2 ) ^{2}} : \frac{6 e ^{3 x}}{( e ^{3 x} + 2 ) ^{2}}

= \frac{6 e ^{3 x}}{( e ^{3 x} + 2 ) ^{2}}

What is the derivative of (e^{3x}/(e^{3x)+2}) ?
The derivative of (e^{3x}/(e^{3x)+2}) is (6e^{3x})/((e^{3x)+2)^2}
What is the first derivative of (e^{3x}/(e^{3x)+2}) ?
The first derivative of (e^{3x}/(e^{3x)+2}) is (6e^{3x})/((e^{3x)+2)^2}