derivative of ln(x+2x) — step-by-step solution

Let's perform the differentiation:

\frac{d}{d x} (\ln (x + 2 x))

ⓘ Like terms are grouped by adding their coefficients:

x + 2 x

3 x

\frac{d}{d x} (\ln (3 x))

Apply the differentiation rule:

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

with f =

3 x

ⓘ The derivative of ln(x) is 1/x:

\frac{d}{d x} (\ln (3 x))

\frac{d}{d x} (3 x) \cdot \frac{1}{3 x}

\frac{d}{d x} (3 x) \cdot \frac{1}{3 x}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} (3 x) \cdot \frac{1}{3 x}

\frac{\frac{d}{d x} (3 x) \cdot 1}{3 x}

\frac{\frac{d}{d x} (3 x)}{3 x}

■A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (3 x)

3

\frac{3}{3 x}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{3}{3 x}

\frac{1}{x}

▶Derivative:

\frac{1}{x}