Calculus Examples

$f (x) = {(10 x^{3} - 6 x)}^{\frac{4}{3}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{4}{3}}$ and $g (x) = 10 x^{3} - 6 x$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $10 x^{3} - 6 x$ .

$\frac{d}{d u} [u^{\frac{4}{3}}] \frac{d}{d x} [10 x^{3} - 6 x]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{4}{3}$ .

$\frac{4}{3} u^{\frac{4}{3} - 1} \frac{d}{d x} [10 x^{3} - 6 x]$

Step 1.3

Replace all occurrences of $u$ with $10 x^{3} - 6 x$ .

$\frac{4}{3} {(10 x^{3} - 6 x)}^{\frac{4}{3} - 1} \frac{d}{d x} [10 x^{3} - 6 x]$

Step 2

By the Sum Rule, the derivative of $10 x^{3} - 6 x$ with respect to $x$ is $\frac{d}{d x} [10 x^{3}] + \frac{d}{d x} [- 6 x]$ .

$\frac{4}{3} {(10 x^{3} - 6 x)}^{\frac{4}{3} - 1} (\frac{d}{d x} [10 x^{3}] + \frac{d}{d x} [- 6 x])$

Step 3

Evaluate $\frac{d}{d x} [10 x^{3}]$ .

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Step 3.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{3}$ with respect to $x$ is $10 \frac{d}{d x} [x^{3}]$ .

$\frac{4}{3} {(10 x^{3} - 6 x)}^{\frac{4}{3} - 1} (10 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x])$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{4}{3} {(10 x^{3} - 6 x)}^{\frac{4}{3} - 1} (10 (3 x^{2}) + \frac{d}{d x} [- 6 x])$

Step 3.3

Multiply $3$ by $10$ .

$\frac{4}{3} {(10 x^{3} - 6 x)}^{\frac{4}{3} - 1} (30 x^{2} + \frac{d}{d x} [- 6 x])$

Step 4

Evaluate $\frac{d}{d x} [- 6 x]$ .

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Step 4.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$\frac{4}{3} {(10 x^{3} - 6 x)}^{\frac{4}{3} - 1} (30 x^{2} - 6 \frac{d}{d x} [x])$

Step 4.2

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

$\frac{4}{3} {(10 x^{3} - 6 x)}^{\frac{4}{3} - 1} (30 x^{2} - 6 \cdot 1)$

Step 4.3

Multiply $- 6$ by $1$ .

$\frac{4}{3} {(10 x^{3} - 6 x)}^{\frac{4}{3} - 1} (30 x^{2} - 6)$

Step 5

Simplify.

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Step 5.1

Combine terms.

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Step 5.1.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4}{3} {(10 x^{3} - 6 x)}^{\frac{4}{3} - 1 \cdot \frac{3}{3}} (30 x^{2} - 6)$

Step 5.1.2

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{4}{3} {(10 x^{3} - 6 x)}^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}} (30 x^{2} - 6)$

Step 5.1.3

Combine the numerators over the common denominator.

$\frac{4}{3} {(10 x^{3} - 6 x)}^{\frac{4 - 1 \cdot 3}{3}} (30 x^{2} - 6)$

Step 5.1.4

Simplify the numerator.

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Step 5.1.4.1

Multiply $- 1$ by $3$ .

$\frac{4}{3} {(10 x^{3} - 6 x)}^{\frac{4 - 3}{3}} (30 x^{2} - 6)$

Step 5.1.4.2

Subtract $3$ from $4$ .

$\frac{4}{3} {(10 x^{3} - 6 x)}^{\frac{1}{3}} (30 x^{2} - 6)$

Step 5.1.5

Combine $\frac{4}{3}$ and ${(10 x^{3} - 6 x)}^{\frac{1}{3}}$ .

$\frac{4 {(10 x^{3} - 6 x)}^{\frac{1}{3}}}{3} (30 x^{2} - 6)$

Step 5.2

Reorder the factors of $\frac{4 {(10 x^{3} - 6 x)}^{\frac{1}{3}}}{3} (30 x^{2} - 6)$ .

$(30 x^{2} - 6) \frac{4 {(10 x^{3} - 6 x)}^{\frac{1}{3}}}{3}$