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

Let's perform the differentiation:

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

We differentiate using:

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

(where f =

\tan (x^{3})

)

ⓘ The derivative of ln(x) is 1/x:

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

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

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

Apply the differentiation rule:

\frac{d}{d x} (\tan (f)) = \frac{d}{d x} (f) \cdot \frac{1}{{\cos (f)}^{2}}

with f =

x^{3}

ⓘ Multiplying a fraction by a term multiplies its numerator:

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

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

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

■The derivative of tan(x) is 1/cos(x)^2:

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

\frac{d}{d x} (x^{3}) \cdot \frac{1}{{\cos (x^{3})}^{2}}

\frac{\frac{d}{d x} (x^{3}) \cdot \frac{1}{{\cos (x^{3})}^{2}}}{\tan (x^{3})}

ⓘ The derivative of x^n is n*x^(n-1):

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

3 x^{2}

\frac{\frac{3 x^{2}}{{\cos (x^{3})}^{2}}}{\tan (x^{3})}

ⓘ Dividing a fraction by

\tan (x^{3})

is multiplying its denominator by

\tan (x^{3})

\frac{\frac{3 x^{2}}{{\cos (x^{3})}^{2}}}{\tan (x^{3})}

\frac{3 x^{2}}{{\cos (x^{3})}^{2} \cdot \tan (x^{3})}

▶Derivative:

\frac{3 x^{2}}{{\cos (x^{3})}^{2} \cdot \tan (x^{3})}