Calculus Examples

$4 (4 x^{3}) (2 - 9 x^{6})$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Multiply $4$ by $4$ .

$\frac{d}{d x} [16 x^{3} (2 - 9 x^{6})]$

Step 1.2

Since $16$ is constant with respect to $x$ , the derivative of $16 x^{3} (2 - 9 x^{6})$ with respect to $x$ is $16 \frac{d}{d x} [x^{3} (2 - 9 x^{6})]$ .

$16 \frac{d}{d x} [x^{3} (2 - 9 x^{6})]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{3}$ and $g (x) = 2 - 9 x^{6}$ .

$16 (x^{3} \frac{d}{d x} [2 - 9 x^{6}] + (2 - 9 x^{6}) \frac{d}{d x} [x^{3}])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $2 - 9 x^{6}$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- 9 x^{6}]$ .

$16 (x^{3} (\frac{d}{d x} [2] + \frac{d}{d x} [- 9 x^{6}]) + (2 - 9 x^{6}) \frac{d}{d x} [x^{3}])$

Step 3.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$16 (x^{3} (0 + \frac{d}{d x} [- 9 x^{6}]) + (2 - 9 x^{6}) \frac{d}{d x} [x^{3}])$

Step 3.3

Add $0$ and $\frac{d}{d x} [- 9 x^{6}]$ .

$16 (x^{3} \frac{d}{d x} [- 9 x^{6}] + (2 - 9 x^{6}) \frac{d}{d x} [x^{3}])$

Step 3.4

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x^{6}$ with respect to $x$ is $- 9 \frac{d}{d x} [x^{6}]$ .

$16 (x^{3} (- 9 \frac{d}{d x} [x^{6}]) + (2 - 9 x^{6}) \frac{d}{d x} [x^{3}])$

Step 3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$16 (x^{3} (- 9 (6 x^{5})) + (2 - 9 x^{6}) \frac{d}{d x} [x^{3}])$

Step 3.6

Multiply $6$ by $- 9$ .

$16 (x^{3} (- 54 x^{5}) + (2 - 9 x^{6}) \frac{d}{d x} [x^{3}])$

Step 4

Multiply $x^{3}$ by $x^{5}$ by adding the exponents.

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Step 4.1

Move $x^{5}$ .

$16 (x^{5} x^{3} \cdot - 54 + (2 - 9 x^{6}) \frac{d}{d x} [x^{3}])$

Step 4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$16 (x^{5 + 3} \cdot - 54 + (2 - 9 x^{6}) \frac{d}{d x} [x^{3}])$

Step 4.3

Add $5$ and $3$ .

$16 (x^{8} \cdot - 54 + (2 - 9 x^{6}) \frac{d}{d x} [x^{3}])$

Step 5

Move $- 54$ to the left of $x^{8}$ .

$16 (- 54 \cdot x^{8} + (2 - 9 x^{6}) \frac{d}{d x} [x^{3}])$

Step 6

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

$16 (- 54 x^{8} + (2 - 9 x^{6}) (3 x^{2}))$

Step 7

Move $3$ to the left of $2 - 9 x^{6}$ .

$16 (- 54 x^{8} + 3 \cdot (2 - 9 x^{6}) x^{2})$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$16 (- 54 x^{8} + (3 \cdot 2 + 3 (- 9 x^{6})) x^{2})$

Step 8.2

Apply the distributive property.

$16 (- 54 x^{8} + 3 \cdot 2 x^{2} + 3 (- 9 x^{6}) x^{2})$

Step 8.3

Apply the distributive property.

$16 (- 54 x^{8}) + 16 (3 \cdot 2 x^{2}) + 16 (3 (- 9 x^{6}) x^{2})$

Step 8.4

Combine terms.

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Step 8.4.1

Multiply $- 54$ by $16$ .

$- 864 x^{8} + 16 (3 \cdot 2 x^{2}) + 16 (3 (- 9 x^{6}) x^{2})$

Step 8.4.2

Multiply $3$ by $2$ .

$- 864 x^{8} + 16 (6 x^{2}) + 16 (3 (- 9 x^{6}) x^{2})$

Step 8.4.3

Multiply $6$ by $16$ .

$- 864 x^{8} + 96 x^{2} + 16 (3 (- 9 x^{6}) x^{2})$

Step 8.4.4

Multiply $- 9$ by $3$ .

$- 864 x^{8} + 96 x^{2} + 16 (- 27 x^{6} x^{2})$

Step 8.4.5

Multiply $x^{6}$ by $x^{2}$ by adding the exponents.

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Step 8.4.5.1

Move $x^{2}$ .

$- 864 x^{8} + 96 x^{2} + 16 (- 27 (x^{2} x^{6}))$

Step 8.4.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 864 x^{8} + 96 x^{2} + 16 (- 27 x^{2 + 6})$

Step 8.4.5.3

Add $2$ and $6$ .

$- 864 x^{8} + 96 x^{2} + 16 (- 27 x^{8})$

Step 8.4.6

Multiply $- 27$ by $16$ .

$- 864 x^{8} + 96 x^{2} - 432 x^{8}$

Step 8.4.7

Subtract $432 x^{8}$ from $- 864 x^{8}$ .

$- 1296 x^{8} + 96 x^{2}$