derivative of ((x+4)/(x-4))^5 — step-by-step solution

1) Let's differentiate:

\frac{d}{d x} ({(\frac{x + 4}{x - 4})}^{5})

We differentiate using:

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

(where f =

\frac{x + 4}{x - 4}

)

5 \cdot \frac{d}{d x} (\frac{x + 4}{x - 4}) \cdot {(\frac{x + 4}{x - 4})}^{4}

We differentiate

u1

x + 4

and

v2

x - 4

using:

\frac{d}{d x} (\frac{u1}{v2}) = \frac{\frac{d}{d x} (u1) \cdot v2 - u1 \cdot \frac{d}{d x} (v2)}{{v2}^{2}}

2) Let's differentiate

u1

\frac{d}{d x} (x + 4)

1 + 0

1

3) Let's perform the differentiation

v2

\frac{d}{d x} (x - 4)

1 + 0

1

4) Let's evaluate the resulting derivative (

\frac{u1}{v2}

)':

\frac{1 \cdot (x - 4) - (x + 4) \cdot 1}{{(x - 4)}^{2}}

\frac{(x - 4) - (x + 4)}{{(x - 4)}^{2}}

\frac{x - 4 - x - 4}{{(x - 4)}^{2}}

ℹ

- 4 - 4

-8

‖

x - x

0

\frac{0 - 8}{{(x - 4)}^{2}}

- \frac{8}{{(x - 4)}^{2}}

5) Then you can continue the derivation:

5 \cdot (- \frac{8}{{(x - 4)}^{2}}) \cdot {(\frac{x + 4}{x - 4})}^{4}

- \frac{40 \cdot {(\frac{x + 4}{x - 4})}^{4}}{{(x - 4)}^{2}}

- \frac{40 \cdot \frac{{(x + 4)}^{4}}{{(x - 4)}^{4}}}{{(x - 4)}^{2}}

- \frac{\frac{40 \cdot {(x + 4)}^{4}}{{(x - 4)}^{4}}}{{(x - 4)}^{2}}

- \frac{40 \cdot {(x + 4)}^{4}}{{(x - 4)}^{4}} \cdot \frac{1}{{(x - 4)}^{2}}

▶Factors identified:

- \frac{40 \cdot {(x + 4)}^{4}}{{(x - 4)}^{6}}