What is the derivative of xln(xy) ?

The derivative of xln(xy) is ln(xy)+(y+x(dy)/(dx))/y

What is the first derivative of xln(xy) ?

The first derivative of xln(xy) is ln(xy)+(y+x(dy)/(dx))/y

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

\frac{d}{d x} (x ln (x y))

ln (x y) + \frac{y + x \frac{d y}{d x}}{y}

Solution steps

\frac{d}{d x} (x ln (x y))

Apply the Product Rule:

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

f = x, g = ln (x y)

= \frac{d x}{d x} ln (x y) + \frac{d}{d x} (ln (x y)) x

\frac{d x}{d x} = 1

\frac{d}{d x} (ln (x y)) = \frac{y + x \frac{d y}{d x}}{x y}

= 1 \cdot ln (x y) + \frac{y + x \frac{d y}{d x}}{x y} x

Simplify

1 \cdot ln (x y) + \frac{y + x \frac{d y}{d x}}{x y} x : ln (x y) + \frac{y + x \frac{d y}{d x}}{y}

= ln (x y) + \frac{y + x \frac{d y}{d x}}{y}

What is the derivative of xln(xy) ?
The derivative of xln(xy) is ln(xy)+(y+x(dy)/(dx))/y
What is the first derivative of xln(xy) ?
The first derivative of xln(xy) is ln(xy)+(y+x(dy)/(dx))/y