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

The derivative of ln((3+e^{2x}/(3-e^{2x)})) is (12e^{2x})/((e^{2x)+3)(-e^{2x}+3)}

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

The first derivative of ln((3+e^{2x}/(3-e^{2x)})) is (12e^{2x})/((e^{2x)+3)(-e^{2x}+3)}

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

\frac{12 e ^{2 x}}{( e ^{2 x} + 3 ) ( - e ^{2 x} + 3 )}

Solution steps

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

Apply the chain rule:

\frac{1}{\frac{3 + e ^{2 x}}{3 - e ^{2 x}}} \frac{d}{d x} (\frac{3 + e ^{2 x}}{3 - e ^{2 x}})

= \frac{1}{\frac{3 + e ^{2 x}}{3 - e ^{2 x}}} \frac{d}{d x} (\frac{3 + e ^{2 x}}{3 - e ^{2 x}})

\frac{d}{d x} (\frac{3 + e ^{2 x}}{3 - e ^{2 x}}) = \frac{12 e ^{2 x}}{( 3 - e ^{2 x} ) ^{2}}

= \frac{1}{\frac{3 + e ^{2 x}}{3 - e ^{2 x}}} \cdot \frac{12 e ^{2 x}}{( 3 - e ^{2 x} ) ^{2}}

Simplify

\frac{1}{\frac{3 + e ^{2 x}}{3 - e ^{2 x}}} \cdot \frac{12 e ^{2 x}}{( 3 - e ^{2 x} ) ^{2}} : \frac{12 e ^{2 x}}{( e ^{2 x} + 3 ) ( - e ^{2 x} + 3 )}

= \frac{12 e ^{2 x}}{( e ^{2 x} + 3 ) ( - e ^{2 x} + 3 )}

What is the derivative of ln((3+e^{2x}/(3-e^{2x)})) ?
The derivative of ln((3+e^{2x}/(3-e^{2x)})) is (12e^{2x})/((e^{2x)+3)(-e^{2x}+3)}
What is the first derivative of ln((3+e^{2x}/(3-e^{2x)})) ?
The first derivative of ln((3+e^{2x}/(3-e^{2x)})) is (12e^{2x})/((e^{2x)+3)(-e^{2x}+3)}