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

Let's differentiate:

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

We differentiate using:

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

(where f =

\tan (x)

)

ⓘ A constant factor stays in front of the derivative, (k*u)' = k*u'. The derivative of x^n is n*x^(n-1):

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

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

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

ⓘ Multiplying by 1 changes nothing:

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

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

ⅰ

\frac{1}{3} \cdot 3

1

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

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

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

\frac{1}{{\cos (x)}^{2}}

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

▶Derivative:

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