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

Let's perform the differentiation:

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

We differentiate using:

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

(where f =

{\ln (x)}^{2}

)

ⓘ The derivative of ln(x) is 1/x:

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

\frac{d}{d x} ({\ln (x)}^{2}) \cdot \frac{1}{{\ln (x)}^{2}}

\frac{d}{d x} ({\ln (x)}^{2}) \cdot \frac{1}{{\ln (x)}^{2}}

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

with f =

\ln (x)

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} ({\ln (x)}^{2}) \cdot \frac{1}{{\ln (x)}^{2}}

\frac{\frac{d}{d x} ({\ln (x)}^{2}) \cdot 1}{{\ln (x)}^{2}}

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

■The derivative of x^n is n*x^(n-1):

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

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

\frac{2 \cdot \frac{d}{d x} (\ln (x)) \cdot \ln (x)}{{\ln (x)}^{2}}

We divide the numerator and denominator of

\frac{\ln (x)}{{\ln (x)}^{2}}

by the common factor:

\ln (x)

, which gives the new fraction:

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

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

ⓘ The derivative of ln(x) is 1/x:

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

\frac{1}{x}

\frac{2 \cdot \frac{1}{x}}{\ln (x)}

ⓘ Multiplying a fraction by a number multiplies its numerator:

2 \cdot \frac{1}{x}

\frac{2 \cdot 1}{x}

\frac{2}{x}

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

ⓘ Dividing a fraction by

\ln (x)

is multiplying its denominator by

\ln (x)

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

\frac{2}{x \cdot \ln (x)}

▶Derivative:

\frac{2}{x \cdot \ln (x)}