Calculus Examples

$y = 6 x^{3} (4 x^{4} - 5 x^{3})$

Step 1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{3} (4 x^{4} - 5 x^{3})$ with respect to $x$ is $6 \frac{d}{d x} [x^{3} (4 x^{4} - 5 x^{3})]$ .

$6 \frac{d}{d x} [x^{3} (4 x^{4} - 5 x^{3})]$

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 $g (x) = 4 x^{4} - 5 x^{3}$ .

$6 (x^{3} \frac{d}{d x} [4 x^{4} - 5 x^{3}] + (4 x^{4} - 5 x^{3}) \frac{d}{d x} [x^{3}])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $4 x^{4} - 5 x^{3}$ with respect to $x$ is $\frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [- 5 x^{3}]$ .

$6 (x^{3} (\frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [- 5 x^{3}]) + (4 x^{4} - 5 x^{3}) \frac{d}{d x} [x^{3}])$

Step 3.2

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

$6 (x^{3} (4 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 5 x^{3}]) + (4 x^{4} - 5 x^{3}) \frac{d}{d x} [x^{3}])$

Step 3.3

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

$6 (x^{3} (4 (4 x^{3}) + \frac{d}{d x} [- 5 x^{3}]) + (4 x^{4} - 5 x^{3}) \frac{d}{d x} [x^{3}])$

Step 3.4

Multiply $4$ by $4$ .

$6 (x^{3} (16 x^{3} + \frac{d}{d x} [- 5 x^{3}]) + (4 x^{4} - 5 x^{3}) \frac{d}{d x} [x^{3}])$

Step 3.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{3}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{3}]$ .

$6 (x^{3} (16 x^{3} - 5 \frac{d}{d x} [x^{3}]) + (4 x^{4} - 5 x^{3}) \frac{d}{d x} [x^{3}])$

Step 3.6

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

$6 (x^{3} (16 x^{3} - 5 (3 x^{2})) + (4 x^{4} - 5 x^{3}) \frac{d}{d x} [x^{3}])$

Step 3.7

Multiply $3$ by $- 5$ .

$6 (x^{3} (16 x^{3} - 15 x^{2}) + (4 x^{4} - 5 x^{3}) \frac{d}{d x} [x^{3}])$

Step 3.8

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

$6 (x^{3} (16 x^{3} - 15 x^{2}) + (4 x^{4} - 5 x^{3}) (3 x^{2}))$

Step 3.9

Move $3$ to the left of $4 x^{4} - 5 x^{3}$ .

$6 (x^{3} (16 x^{3} - 15 x^{2}) + 3 \cdot (4 x^{4} - 5 x^{3}) x^{2})$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$6 (x^{3} (16 x^{3}) + x^{3} (- 15 x^{2}) + 3 (4 x^{4} - 5 x^{3}) x^{2})$

Step 4.2

Apply the distributive property.

$6 (x^{3} (16 x^{3}) + x^{3} (- 15 x^{2}) + (3 (4 x^{4}) + 3 (- 5 x^{3})) x^{2})$

Step 4.3

Apply the distributive property.

$6 (x^{3} (16 x^{3}) + x^{3} (- 15 x^{2}) + 3 (4 x^{4}) x^{2} + 3 (- 5 x^{3}) x^{2})$

Step 4.4

Apply the distributive property.

$6 (x^{3} (16 x^{3})) + 6 (x^{3} (- 15 x^{2})) + 6 (3 (4 x^{4}) x^{2}) + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5

Combine terms.

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Step 4.5.1

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

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Step 4.5.1.1

Move $x^{3}$ .

$6 (x^{3} x^{3} \cdot 16) + 6 (x^{3} (- 15 x^{2})) + 6 (3 (4 x^{4}) x^{2}) + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5.1.2

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

$6 (x^{3 + 3} \cdot 16) + 6 (x^{3} (- 15 x^{2})) + 6 (3 (4 x^{4}) x^{2}) + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5.1.3

Add $3$ and $3$ .

$6 (x^{6} \cdot 16) + 6 (x^{3} (- 15 x^{2})) + 6 (3 (4 x^{4}) x^{2}) + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5.2

Move $16$ to the left of $x^{6}$ .

$6 (16 \cdot x^{6}) + 6 (x^{3} (- 15 x^{2})) + 6 (3 (4 x^{4}) x^{2}) + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5.3

Multiply $16$ by $6$ .

$96 x^{6} + 6 (x^{3} (- 15 x^{2})) + 6 (3 (4 x^{4}) x^{2}) + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5.4

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 4.5.4.1

Move $x^{2}$ .

$96 x^{6} + 6 (x^{2} x^{3} \cdot - 15) + 6 (3 (4 x^{4}) x^{2}) + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5.4.2

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

$96 x^{6} + 6 (x^{2 + 3} \cdot - 15) + 6 (3 (4 x^{4}) x^{2}) + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5.4.3

Add $2$ and $3$ .

$96 x^{6} + 6 (x^{5} \cdot - 15) + 6 (3 (4 x^{4}) x^{2}) + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5.5

Move $- 15$ to the left of $x^{5}$ .

$96 x^{6} + 6 (- 15 \cdot x^{5}) + 6 (3 (4 x^{4}) x^{2}) + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5.6

Multiply $- 15$ by $6$ .

$96 x^{6} - 90 x^{5} + 6 (3 (4 x^{4}) x^{2}) + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5.7

Multiply $4$ by $3$ .

$96 x^{6} - 90 x^{5} + 6 (12 x^{4} x^{2}) + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5.8

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 4.5.8.1

Move $x^{2}$ .

$96 x^{6} - 90 x^{5} + 6 (12 (x^{2} x^{4})) + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5.8.2

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

$96 x^{6} - 90 x^{5} + 6 (12 x^{2 + 4}) + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5.8.3

Add $2$ and $4$ .

$96 x^{6} - 90 x^{5} + 6 (12 x^{6}) + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5.9

Multiply $12$ by $6$ .

$96 x^{6} - 90 x^{5} + 72 x^{6} + 6 (3 (- 5 x^{3}) x^{2})$

Step 4.5.10

Multiply $- 5$ by $3$ .

$96 x^{6} - 90 x^{5} + 72 x^{6} + 6 (- 15 x^{3} x^{2})$

Step 4.5.11

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 4.5.11.1

Move $x^{2}$ .

$96 x^{6} - 90 x^{5} + 72 x^{6} + 6 (- 15 (x^{2} x^{3}))$

Step 4.5.11.2

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

$96 x^{6} - 90 x^{5} + 72 x^{6} + 6 (- 15 x^{2 + 3})$

Step 4.5.11.3

Add $2$ and $3$ .

$96 x^{6} - 90 x^{5} + 72 x^{6} + 6 (- 15 x^{5})$

Step 4.5.12

Multiply $- 15$ by $6$ .

$96 x^{6} - 90 x^{5} + 72 x^{6} - 90 x^{5}$

Step 4.5.13

Add $96 x^{6}$ and $72 x^{6}$ .

$168 x^{6} - 90 x^{5} - 90 x^{5}$

Step 4.5.14

Subtract $90 x^{5}$ from $- 90 x^{5}$ .

$168 x^{6} - 180 x^{5}$