What is the derivative of ln((a^x/(1+a^x))) ?

The derivative of ln((a^x/(1+a^x))) is (ln(a))/(1+a^x)

What is the first derivative of ln((a^x/(1+a^x))) ?

The first derivative of ln((a^x/(1+a^x))) is (ln(a))/(1+a^x)

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\frac{d}{d x} (ln (\frac{a ^{x}}{1 + a ^{x}}))

\frac{ln ( a )}{1 + a ^{x}}

Solution steps

\frac{d}{d x} (ln (\frac{a ^{x}}{1 + a ^{x}}))

Apply the chain rule:

\frac{1}{\frac{a ^{x}}{1 + a ^{x}}} \frac{d}{d x} (\frac{a ^{x}}{1 + a ^{x}})

= \frac{1}{\frac{a ^{x}}{1 + a ^{x}}} \frac{d}{d x} (\frac{a ^{x}}{1 + a ^{x}})

\frac{d}{d x} (\frac{a ^{x}}{1 + a ^{x}}) = \frac{a ^{x} ln ( a )}{( 1 + a ^{x} ) ^{2}}

= \frac{1}{\frac{a ^{x}}{1 + a ^{x}}} \cdot \frac{a ^{x} ln ( a )}{( 1 + a ^{x} ) ^{2}}

Simplify

\frac{1}{\frac{a ^{x}}{1 + a ^{x}}} \cdot \frac{a ^{x} ln ( a )}{( 1 + a ^{x} ) ^{2}} : \frac{ln ( a )}{1 + a ^{x}}

= \frac{ln ( a )}{1 + a ^{x}}

What is the derivative of ln((a^x/(1+a^x))) ?
The derivative of ln((a^x/(1+a^x))) is (ln(a))/(1+a^x)
What is the first derivative of ln((a^x/(1+a^x))) ?
The first derivative of ln((a^x/(1+a^x))) is (ln(a))/(1+a^x)