What is the derivative of x*ln(x+1) ?

The derivative of x*ln(x+1) is ln(x+1)+x/(x+1)

What is the first derivative of x*ln(x+1) ?

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

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

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

Solution steps

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

Apply the Product Rule:

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

f = x, g = ln (x + 1)

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

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

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

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

Simplify

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

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

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