en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

derivative of ln(sqrt(1+e^x))

Solution

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

Solution

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

Popular Examples

derivative of 4(x^3-1^3)

\frac{d}{d x} (4 (x^{3} - 1)^{3})

limit as x approaches a of 4/(sqrt(1-x))

x \to a lim (\frac{4}{1 - x})

f(x)=sqrt(ln(x))

f (x) = ln (x)

integral of (2*3^x)/(2^x)

\int \frac{2 \cdot 3 ^{x}}{2 ^{x}} d x

integral from 0 to pi of tan(x/3)

\int_{0}^{π} tan (\frac{x}{3}) d x

Frequently Asked Questions (FAQ)

What is the derivative of ln(sqrt(1+e^x)) ?
The derivative of ln(sqrt(1+e^x)) is (e^x)/(2(1+e^x))
What is the first derivative of ln(sqrt(1+e^x)) ?
The first derivative of ln(sqrt(1+e^x)) is (e^x)/(2(1+e^x))