derivative of-2/(9(1+x)^{5/3)} — step-by-step solution

To ease differentiation, the expression

- \frac{2}{9 \cdot {(1 + x)}^{\frac{5}{3}}}

is turned into

- \frac{2}{9} \cdot {(1 + x)}^{- \frac{5}{3}}

Let's differentiate:

\frac{d}{d x} (- \frac{2}{9} \cdot {(1 + x)}^{- \frac{5}{3}})

Apply the differentiation rule:

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

with f =

1 + x

- \frac{2}{9} \cdot (- \frac{5}{3} \cdot \frac{d}{d x} (1 + x) \cdot {(1 + x)}^{(- \frac{5}{3} - 1)})

ℹ

\frac{d}{d x} (1 + x)

0 + 1

‖

- \frac{5}{3} - 1

\frac{- 5 - 1 \cdot 3}{3}

- \frac{2}{9} \cdot (- \frac{5}{3}) \cdot (0 + 1) \cdot {(1 + x)}^{(\frac{- 5 - 1 \cdot 3}{3})}

- \frac{2}{9} \cdot (- \frac{5}{3}) \cdot {(1 + x)}^{(\frac{- 5 - 1 \cdot 3}{3})}

ℹ

-1 \cdot 3

-3

‖

- \frac{2}{9} \cdot (- \frac{5}{3})

\frac{10}{27}

\frac{10}{27} \cdot {(1 + x)}^{(\frac{- 5 - 3}{3})}

ℹ

\frac{10}{27} \cdot {(1 + x)}^{(\frac{- 5 - 3}{3})}

\frac{10 \cdot {(1 + x)}^{(\frac{- 5 - 3}{3})}}{27}

‖

- 5 - 3

-8

\frac{10 \cdot {(1 + x)}^{- \frac{8}{3}}}{27}

\frac{10}{{(1 + x)}^{\frac{8}{3}} \cdot 27}

▶Derivative:

\frac{10}{27 \cdot {(x + 1)}^{\frac{8}{3}}}