Find the Derivative - d/dx (x+8)^(4/5)

${(x + 8)}^{\frac{4}{5}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{4}{5}}$ and $g (x) = x + 8$ .

To apply the Chain Rule, set $u$ as $x + 8$ .

$\frac{d}{d u} [u^{\frac{4}{5}}] \frac{d}{d x} [x + 8]$

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{4}{5}$ .

$\frac{4}{5} u^{\frac{4}{5} - 1} \frac{d}{d x} [x + 8]$

Replace all occurrences of $u$ with $x + 8$ .

$\frac{4}{5} {(x + 8)}^{\frac{4}{5} - 1} \frac{d}{d x} [x + 8]$

Step 2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4}{5} {(x + 8)}^{\frac{4}{5} - 1 \cdot \frac{5}{5}} \frac{d}{d x} [x + 8]$

Step 3

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{4}{5} {(x + 8)}^{\frac{4}{5} + \frac{- 1 \cdot 5}{5}} \frac{d}{d x} [x + 8]$

Step 4

Combine the numerators over the common denominator.

$\frac{4}{5} {(x + 8)}^{\frac{4 - 1 \cdot 5}{5}} \frac{d}{d x} [x + 8]$

Step 5

Simplify the numerator.

Multiply $- 1$ by $5$ .

$\frac{4}{5} {(x + 8)}^{\frac{4 - 5}{5}} \frac{d}{d x} [x + 8]$

Subtract $5$ from $4$ .

$\frac{4}{5} {(x + 8)}^{\frac{- 1}{5}} \frac{d}{d x} [x + 8]$

Step 6

Combine fractions.

Move the negative in front of the fraction.

$\frac{4}{5} {(x + 8)}^{- \frac{1}{5}} \frac{d}{d x} [x + 8]$

Combine $\frac{4}{5}$ and ${(x + 8)}^{- \frac{1}{5}}$ .

$\frac{4 {(x + 8)}^{- \frac{1}{5}}}{5} \frac{d}{d x} [x + 8]$

Move ${(x + 8)}^{- \frac{1}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4}{5 {(x + 8)}^{\frac{1}{5}}} \frac{d}{d x} [x + 8]$

Step 7

By the Sum Rule, the derivative of $x + 8$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [8]$ .

$\frac{4}{5 {(x + 8)}^{\frac{1}{5}}} (\frac{d}{d x} [x] + \frac{d}{d x} [8])$

Step 8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{4}{5 {(x + 8)}^{\frac{1}{5}}} (1 + \frac{d}{d x} [8])$

Step 9

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{4}{5 {(x + 8)}^{\frac{1}{5}}} (1 + 0)$

Step 10

Simplify the expression.

Add $1$ and $0$ .

$\frac{4}{5 {(x + 8)}^{\frac{1}{5}}} \cdot 1$

Multiply $\frac{4}{5 {(x + 8)}^{\frac{1}{5}}}$ by $1$ .

$\frac{4}{5 {(x + 8)}^{\frac{1}{5}}}$

Find the Derivative - d/dx (x+8)^(4/5)

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