derivative of ln(4x^3) — step-by-step solution

Let's perform the differentiation:

\frac{d}{d x} (\ln (4 x^{3}))

We differentiate using:

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

(where f =

4 x^{3}

)

ⓘ The derivative of ln(x) is 1/x:

\frac{d}{d x} (\ln (4 x^{3}))

\frac{d}{d x} (4 x^{3}) \cdot \frac{1}{4 x^{3}}

\frac{d}{d x} (4 x^{3}) \cdot \frac{1}{4 x^{3}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} (4 x^{3}) \cdot \frac{1}{4 x^{3}}

\frac{\frac{d}{d x} (4 x^{3}) \cdot 1}{4 x^{3}}

\frac{\frac{d}{d x} (4 x^{3})}{4 x^{3}}

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

\frac{d}{d x} (4 x^{3})

4 \cdot \frac{d}{d x} (x^{3})

■The derivative of x^n is n*x^(n-1):

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

3 x^{2}

\frac{4 \cdot 3 x^{2}}{4 x^{3}}

ⅰ

4 \cdot 3

12

\frac{12 x^{2}}{4 x^{3}}

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

\frac{12 x^{2}}{4 x^{3}}

\frac{12}{4 x}

\frac{12}{4 x}

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

\frac{12}{4 x}

\frac{3}{x}

▶Derivative:

\frac{3}{x}