Calculus Examples

$G (x) = 9 x^{5} (x^{4} - 9)$

Step 1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{5} (x^{4} - 9)$ with respect to $x$ is $9 \frac{d}{d x} [x^{5} (x^{4} - 9)]$ .

$9 \frac{d}{d x} [x^{5} (x^{4} - 9)]$

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 $f (x) = x^{5}$ and $g (x) = x^{4} - 9$ .

$9 (x^{5} \frac{d}{d x} [x^{4} - 9] + (x^{4} - 9) \frac{d}{d x} [x^{5}])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x^{4} - 9$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 9]$ .

$9 (x^{5} (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 9]) + (x^{4} - 9) \frac{d}{d x} [x^{5}])$

Step 3.2

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

$9 (x^{5} (4 x^{3} + \frac{d}{d x} [- 9]) + (x^{4} - 9) \frac{d}{d x} [x^{5}])$

Step 3.3

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$9 (x^{5} (4 x^{3} + 0) + (x^{4} - 9) \frac{d}{d x} [x^{5}])$

Step 3.4

Add $4 x^{3}$ and $0$ .

$9 (x^{5} (4 x^{3}) + (x^{4} - 9) \frac{d}{d x} [x^{5}])$

Step 4

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

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Step 4.1

Move $x^{3}$ .

$9 (x^{3} x^{5} \cdot 4 + (x^{4} - 9) \frac{d}{d x} [x^{5}])$

Step 4.2

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

$9 (x^{3 + 5} \cdot 4 + (x^{4} - 9) \frac{d}{d x} [x^{5}])$

Step 4.3

Add $3$ and $5$ .

$9 (x^{8} \cdot 4 + (x^{4} - 9) \frac{d}{d x} [x^{5}])$

Step 5

Move $4$ to the left of $x^{8}$ .

$9 (4 \cdot x^{8} + (x^{4} - 9) \frac{d}{d x} [x^{5}])$

Step 6

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

$9 (4 x^{8} + (x^{4} - 9) (5 x^{4}))$

Step 7

Move $5$ to the left of $x^{4} - 9$ .

$9 (4 x^{8} + 5 \cdot (x^{4} - 9) x^{4})$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$9 (4 x^{8} + (5 x^{4} + 5 \cdot - 9) x^{4})$

Step 8.2

Apply the distributive property.

$9 (4 x^{8} + 5 x^{4} x^{4} + 5 \cdot - 9 x^{4})$

Step 8.3

Apply the distributive property.

$9 (4 x^{8}) + 9 (5 x^{4} x^{4}) + 9 (5 \cdot - 9 x^{4})$

Step 8.4

Combine terms.

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Step 8.4.1

Multiply $4$ by $9$ .

$36 x^{8} + 9 (5 x^{4} x^{4}) + 9 (5 \cdot - 9 x^{4})$

Step 8.4.2

Multiply $x^{4}$ by $x^{4}$ by adding the exponents.

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Step 8.4.2.1

Move $x^{4}$ .

$36 x^{8} + 9 (5 (x^{4} x^{4})) + 9 (5 \cdot - 9 x^{4})$

Step 8.4.2.2

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

$36 x^{8} + 9 (5 x^{4 + 4}) + 9 (5 \cdot - 9 x^{4})$

Step 8.4.2.3

Add $4$ and $4$ .

$36 x^{8} + 9 (5 x^{8}) + 9 (5 \cdot - 9 x^{4})$

Step 8.4.3

Multiply $5$ by $9$ .

$36 x^{8} + 45 x^{8} + 9 (5 \cdot - 9 x^{4})$

Step 8.4.4

Multiply $5$ by $- 9$ .

$36 x^{8} + 45 x^{8} + 9 (- 45 x^{4})$

Step 8.4.5

Multiply $- 45$ by $9$ .

$36 x^{8} + 45 x^{8} - 405 x^{4}$

Step 8.4.6

Add $36 x^{8}$ and $45 x^{8}$ .

$81 x^{8} - 405 x^{4}$