differentiate x*ln(x) — step-by-step solution

Let's perform the differentiation:

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

For

u1

x

and

v2

\ln (x)

, use the differentiation rule:

\frac{d}{d x} (u1 \cdot v2) = \frac{d}{d x} (u1) \cdot v2 + u1 \cdot \frac{d}{d x} (v2)

Let's differentiate

u1

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

ⓘ The derivative of x is 1:

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

1

1

Let's perform the differentiation

v2

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

ⓘ The derivative of ln(x) is 1/x:

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

\frac{1}{x}

\frac{1}{x}

Let's evaluate the resulting derivative (

u1 \cdot v2

)':

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

ⓘ Multiplying a fraction by a term multiplies its numerator:

x \cdot \frac{1}{x}

\frac{x \cdot 1}{x}

\frac{x}{x}

■The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor

x

\frac{x}{x}

1

ⅰ

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

\ln (x) + 1

▶Derivative:

\ln (x) + 1