Calculus Examples

$2 x^{4} (3 x^{2} + 7 x)$

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{4} (3 x^{2} + 7 x)$ with respect to $x$ is $2 \frac{d}{d x} [x^{4} (3 x^{2} + 7 x)]$ .

$2 \frac{d}{d x} [x^{4} (3 x^{2} + 7 x)]$

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 $f (x) = x^{4}$ and $g (x) = 3 x^{2} + 7 x$ .

$2 (x^{4} \frac{d}{d x} [3 x^{2} + 7 x] + (3 x^{2} + 7 x) \frac{d}{d x} [x^{4}])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $3 x^{2} + 7 x$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [7 x]$ .

$2 (x^{4} (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [7 x]) + (3 x^{2} + 7 x) \frac{d}{d x} [x^{4}])$

Step 3.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$2 (x^{4} (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [7 x]) + (3 x^{2} + 7 x) \frac{d}{d x} [x^{4}])$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 (x^{4} (3 (2 x) + \frac{d}{d x} [7 x]) + (3 x^{2} + 7 x) \frac{d}{d x} [x^{4}])$

Step 3.4

Multiply $2$ by $3$ .

$2 (x^{4} (6 x + \frac{d}{d x} [7 x]) + (3 x^{2} + 7 x) \frac{d}{d x} [x^{4}])$

Step 3.5

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$2 (x^{4} (6 x + 7 \frac{d}{d x} [x]) + (3 x^{2} + 7 x) \frac{d}{d x} [x^{4}])$

Step 3.6

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

$2 (x^{4} (6 x + 7 \cdot 1) + (3 x^{2} + 7 x) \frac{d}{d x} [x^{4}])$

Step 3.7

Multiply $7$ by $1$ .

$2 (x^{4} (6 x + 7) + (3 x^{2} + 7 x) \frac{d}{d x} [x^{4}])$

Step 3.8

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

$2 (x^{4} (6 x + 7) + (3 x^{2} + 7 x) (4 x^{3}))$

Step 3.9

Move $4$ to the left of $3 x^{2} + 7 x$ .

$2 (x^{4} (6 x + 7) + 4 \cdot (3 x^{2} + 7 x) x^{3})$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$2 (x^{4} (6 x) + x^{4} \cdot 7 + 4 (3 x^{2} + 7 x) x^{3})$

Step 4.2

Apply the distributive property.

$2 (x^{4} (6 x) + x^{4} \cdot 7 + (4 (3 x^{2}) + 4 (7 x)) x^{3})$

Step 4.3

Apply the distributive property.

$2 (x^{4} (6 x) + x^{4} \cdot 7 + 4 (3 x^{2}) x^{3} + 4 (7 x) x^{3})$

Step 4.4

Apply the distributive property.

$2 (x^{4} (6 x)) + 2 (x^{4} \cdot 7) + 2 (4 (3 x^{2}) x^{3}) + 2 (4 (7 x) x^{3})$

Step 4.5

Combine terms.

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Step 4.5.1

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

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Step 4.5.1.1

Move $x$ .

$2 (x \cdot x^{4} \cdot 6) + 2 (x^{4} \cdot 7) + 2 (4 (3 x^{2}) x^{3}) + 2 (4 (7 x) x^{3})$

Step 4.5.1.2

Multiply $x$ by $x^{4}$ .

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Step 4.5.1.2.1

Raise $x$ to the power of $1$ .

$2 (x^{1} x^{4} \cdot 6) + 2 (x^{4} \cdot 7) + 2 (4 (3 x^{2}) x^{3}) + 2 (4 (7 x) x^{3})$

Step 4.5.1.2.2

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

$2 (x^{1 + 4} \cdot 6) + 2 (x^{4} \cdot 7) + 2 (4 (3 x^{2}) x^{3}) + 2 (4 (7 x) x^{3})$

Step 4.5.1.3

Add $1$ and $4$ .

$2 (x^{5} \cdot 6) + 2 (x^{4} \cdot 7) + 2 (4 (3 x^{2}) x^{3}) + 2 (4 (7 x) x^{3})$

Step 4.5.2

Move $6$ to the left of $x^{5}$ .

$2 (6 \cdot x^{5}) + 2 (x^{4} \cdot 7) + 2 (4 (3 x^{2}) x^{3}) + 2 (4 (7 x) x^{3})$

Step 4.5.3

Multiply $6$ by $2$ .

$12 x^{5} + 2 (x^{4} \cdot 7) + 2 (4 (3 x^{2}) x^{3}) + 2 (4 (7 x) x^{3})$

Step 4.5.4

Move $7$ to the left of $x^{4}$ .

$12 x^{5} + 2 (7 \cdot x^{4}) + 2 (4 (3 x^{2}) x^{3}) + 2 (4 (7 x) x^{3})$

Step 4.5.5

Multiply $7$ by $2$ .

$12 x^{5} + 14 x^{4} + 2 (4 (3 x^{2}) x^{3}) + 2 (4 (7 x) x^{3})$

Step 4.5.6

Multiply $3$ by $4$ .

$12 x^{5} + 14 x^{4} + 2 (12 x^{2} x^{3}) + 2 (4 (7 x) x^{3})$

Step 4.5.7

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

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Step 4.5.7.1

Move $x^{3}$ .

$12 x^{5} + 14 x^{4} + 2 (12 (x^{3} x^{2})) + 2 (4 (7 x) x^{3})$

Step 4.5.7.2

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

$12 x^{5} + 14 x^{4} + 2 (12 x^{3 + 2}) + 2 (4 (7 x) x^{3})$

Step 4.5.7.3

Add $3$ and $2$ .

$12 x^{5} + 14 x^{4} + 2 (12 x^{5}) + 2 (4 (7 x) x^{3})$

Step 4.5.8

Multiply $12$ by $2$ .

$12 x^{5} + 14 x^{4} + 24 x^{5} + 2 (4 (7 x) x^{3})$

Step 4.5.9

Multiply $7$ by $4$ .

$12 x^{5} + 14 x^{4} + 24 x^{5} + 2 (28 x \cdot x^{3})$

Step 4.5.10

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

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Step 4.5.10.1

Move $x^{3}$ .

$12 x^{5} + 14 x^{4} + 24 x^{5} + 2 (28 (x^{3} x))$

Step 4.5.10.2

Multiply $x^{3}$ by $x$ .

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Step 4.5.10.2.1

Raise $x$ to the power of $1$ .

$12 x^{5} + 14 x^{4} + 24 x^{5} + 2 (28 (x^{3} x^{1}))$

Step 4.5.10.2.2

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

$12 x^{5} + 14 x^{4} + 24 x^{5} + 2 (28 x^{3 + 1})$

Step 4.5.10.3

Add $3$ and $1$ .

$12 x^{5} + 14 x^{4} + 24 x^{5} + 2 (28 x^{4})$

Step 4.5.11

Multiply $28$ by $2$ .

$12 x^{5} + 14 x^{4} + 24 x^{5} + 56 x^{4}$

Step 4.5.12

Add $12 x^{5}$ and $24 x^{5}$ .

$36 x^{5} + 14 x^{4} + 56 x^{4}$

Step 4.5.13

Add $14 x^{4}$ and $56 x^{4}$ .

$36 x^{5} + 70 x^{4}$