What is the derivative of (x-5*ln(x/5)) ?

The derivative of (x-5*ln(x/5)) is ln(x/5)+(x-5)/x

What is the first derivative of (x-5*ln(x/5)) ?

The first derivative of (x-5*ln(x/5)) is ln(x/5)+(x-5)/x

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

\frac{d}{d x} ((x - 5) \cdot ln (\frac{x}{5}))

ln (\frac{x}{5}) + \frac{x - 5}{x}

Solution steps

\frac{d}{d x} ((x - 5) ln (\frac{x}{5}))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x - 5, g = ln (\frac{x}{5})

= \frac{d}{d x} (x - 5) ln (\frac{x}{5}) + \frac{d}{d x} (ln (\frac{x}{5})) (x - 5)

\frac{d}{d x} (x - 5) = 1

\frac{d}{d x} (ln (\frac{x}{5})) = \frac{1}{x}

= 1 \cdot ln (\frac{x}{5}) + \frac{1}{x} (x - 5)

Simplify

1 \cdot ln (\frac{x}{5}) + \frac{1}{x} (x - 5) : ln (\frac{x}{5}) + \frac{x - 5}{x}

= ln (\frac{x}{5}) + \frac{x - 5}{x}

What is the derivative of (x-5*ln(x/5)) ?
The derivative of (x-5*ln(x/5)) is ln(x/5)+(x-5)/x
What is the first derivative of (x-5*ln(x/5)) ?
The first derivative of (x-5*ln(x/5)) is ln(x/5)+(x-5)/x