Calculus Examples

$y = {(x^{5} - 8)}^{\frac{4}{5}}$

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 $g (x) = x^{5} - 8$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{5} - 8$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $x^{5} - 8$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $5$ .

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

Step 5.2

Subtract $5$ from $4$ .

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

Step 6

Combine fractions.

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Step 6.1

Move the negative in front of the fraction.

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

By the Sum Rule, the derivative of $x^{5} - 8$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 8]$ .

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify terms.

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Step 10.1

Add $5 x^{4}$ and $0$ .

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

Step 10.2

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

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

Step 10.3

Multiply $5$ by $4$ .

$\frac{20}{5 {(x^{5} - 8)}^{\frac{1}{5}}} x^{4}$

Step 10.4

Combine $\frac{20}{5 {(x^{5} - 8)}^{\frac{1}{5}}}$ and $x^{4}$ .

$\frac{20 x^{4}}{5 {(x^{5} - 8)}^{\frac{1}{5}}}$

Step 10.5

Factor $5$ out of $20 x^{4}$ .

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

Step 11

Cancel the common factors.

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Step 11.1

Factor $5$ out of $5 {(x^{5} - 8)}^{\frac{1}{5}}$ .

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

Step 11.2

Cancel the common factor.

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

Step 11.3

Rewrite the expression.

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