What is (\partial)/(\partial x)(ln(1/(2+3e^x))) ?

The answer to (\partial)/(\partial x)(ln(1/(2+3e^x))) is -(3e^x)/(2+3e^x)

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(\partial)/(\partial x)(ln(1/(2+3e^x)))

\frac{\partial}{\partial x} (ln (\frac{1}{2 + 3 e ^{x}}))

- \frac{3 e ^{x}}{2 + 3 e ^{x}}

Solution steps

\frac{\partial}{\partial x} (ln (\frac{1}{2 + 3 e ^{x}}))

Simplify

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

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

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

x \to 1 lim (\frac{3 x - 2}{x - 1})

\frac{1}{3} tan^{3} (x) - tan (x) + x

x \to - 8 lim (\frac{x + 2}{x + 8})

x \to 2 lim (2 x^{2} + 7 x - 1)

x \to - 3 + lim (2 x^{2} - 6)

What is (\partial)/(\partial x)(ln(1/(2+3e^x))) ?
The answer to (\partial)/(\partial x)(ln(1/(2+3e^x))) is -(3e^x)/(2+3e^x)